Price and risk evaluation system for financial product or its derivatives, dealing system, recording medium storing a price and risk evaluation program, and recording medium storing a dealing program

ABSTRACT

A system for correctly evaluating a price distribution and a risk distribution for a financial product or its derivatives introduces a probability density function generated with a Boltzmann model at a higher accuracy than the Gaussian distribution for a probability density. The system has an initial value setup unit and an evaluation condition setup unit. Initial values include at least one of price, price change rate, and the price change direction of a financial product. The evaluation conditions include at least time steps and the number of trials. The Boltzmann model analysis unit receives the initial values and the evaluation conditions, and repeats simulations of price fluctuation, based on the Boltzmann model using a Monte Carlo method. A velocity/direction distribution setup unit supplies the probability distributions of the price, price change rate, and the price change direction for the financial product to the Boltzmann model analysis unit. A random number generator for a Monte Carlo method employed in the analysis by the Boltzmann model, and an output unit displays the analysis result. A dealing system applies the financial Boltzmann model to option pricing, and reproduces the characteristics of Leptokurcity and Fat-tail by linear Boltzmann equation in order to define risk-neutral and unique probability measures. Consequently, option prices can be evaluated in a risk-neutral and unique manner, taking into account Leptokurcity and Fat-tail of a price change distribution.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a Division of and claims priority under 35 USC § 120from application Ser. No. 09/807,963 filed Jun. 1, 2001, and is aNational Stage of PCT Application No. PCT/JP00/05755, filed Aug. 25,2000, and claims the benefit of priority under 35 USC § 119 fromJapanese Patent Application Nos. P11-242152, filed Aug. 27, 1999 andP2000-219655, filed Jul. 19, 2000, the entire contents of each areincorporated herein by reference.

FIELD OF THE INVENTION

The present invention relates to a system for assessing a pricedistribution or a risk distribution for a financial product or itsderivatives, which can rigorously evaluate a price distribution or arisk distribution, including a probability of occurrence of a big pricechange, based on a Boltzmann model. This system is also capable ofanalyzing price fluctuation events for the financial product or itsderivatives that could not be reproduced by the conventional technique.

The present invention also relates to a dealing system used in thefinancial field.

The present invention further relates to a computer-readable recordingmedium storing a price and risk assessing program for a financialproduct or its derivatives, and to a computer-readable recording mediumstoring a dealing program.

TECHNOLOGICAL BACKGROUND

A technique for analyzing past records of price change in a financialproduct or its derivatives and for stochastically obtaining a pricedistribution or a risk distribution is generally called a financialengineering technology.

In general, the Wiener process is used to model a change of stock pricein the conventional financial engineering technology. The Wiener processis a type of the Markov process, which is a stochastic process oncondition that a future state is independent of a past process. TheWiener process is often used to describe the Brownian motions ofgas-molecules.

With variables of t (time) and z depending on the Wiener process, theWiener process is characterized in the following relationship between Δtand Δz that is an infinitesimal change in z during the infinitesimaltime Δt.ΔZ=ε√{square root over (Δt)}  (1)here ε is the random sample from the standard Gaussian distribution.

Thus, the Wiener process evaluates fluctuations with the variables basedon the standard Gaussian distribution.

The conventional risk evaluation method for a financial product or itsderivatives generally establishes upon applying the Ito's process, whichis developed from the Wiener process. The Ito's process adds a driftterm to the Wiener process on the assumption that a change of stockprice follows the Wiener process, and further introduces a parameterfunction of time and other variables.

The price change in stock priced expressed by the Ito's process isdS=rSdt+Sσ√{square root over (dt)}W  (2)here S is the stock price, r is the non-risky interest rate, σ is thevolatility (i.e., the predicted change rate), and W is the normaldistribution with the expectation value of zero (0) and the standarddeviation of one (1).

The simplest example of the Ito's process is the geometric Brownianmotion model of stock prices. With the geometric Brownian motion,equation (2) becomes $\begin{matrix}{{dx} = {{( {r - \frac{\sigma^{2}}{2}} ){dt}} + {\sigma\sqrt{dt}W}}} & (3)\end{matrix}$here x is the natural logarithm of the stock price S.

The probability density function P(x; t) of x based on equation (3) is$\begin{matrix}{\frac{\partial P}{\partial t} = {{{- ( {r - \frac{\sigma^{2}}{2}} )}\frac{\partial P}{\partial x}} + {\frac{1}{2}\sigma^{2}\frac{\partial^{2}P}{\partial x^{2}}}}} & (4)\end{matrix}$

Equation 4 is the Fokker-Plank equation, and is a typical diffusionproblem. The solution of equation (4) is $\begin{matrix}{{P( {x;t} )} = {\frac{1}{\sqrt{2{\pi\sigma}^{2}t}}{\exp\lbrack {- \frac{( {x - {( {r - \frac{\sigma^{2}}{2}} )t^{2}}} )}{2\sigma^{2}t}} \rbrack}}} & (5)\end{matrix}$and the probability density function P(x; t) of x becomes the Gaussiandistribution.

Equation (5) is characterized by not only its simple form, but alsoeffectiveness in evaluating price changes for financial products,because it is known that the price change of derivatives derived fromthe underlying assets has the same shape as that of the underlyingassets (Ito's theorem). For this reason, various financial derivativeshave been reproduced.

However, the conventional technique for evaluating risks for thefinancial products or the derivatives is not capable of providingsufficiently reliable results, as is known in this field.

This is because that conventional method evaluates risks of financialproducts based on the Gaussian distribution, and therefore, theprobability of occurrence of a big price change is underestimated.

Although the likelihood of occurrence of a big price change is low, sucha big price change has a significant influence to investing risks, ascompared with situations under the normal price changes. Accordingly,any risk evaluation methods or systems for financial products can not bereliable in the practical aspect unless the probability for the bigprice changes is accurately treated.

Another problem is that the conventional risk evaluation techniquerequires some corrections to a heterogeneous problem, in which theprobability density function changes depending on prices, or to anon-linear problem, in which the probability density function used forthe evaluation is a non-linear function. Along with the conventionalapproaches, such corrections have to be added empirically or based onknow-how. In other words, the conventional technique requires dealer'sexperiences or uncertain judgements in the actual market trading.

Furthermore, the conventional risk evaluation technique has very limitedcapabilities for description, definition, and selection of the variablesto produce price fluctuations observed in the markets. In other words,with the conventional technique, the probability density function cannot be sufficiently evaluated with variables for describing risks forfinancial products, if the actual price change distribution of anfinancial product is located out of the standard Gaussian-typedistribution. This insufficiency can also be true in the cases where theprice change rate is influenced by the past price change rate, andcorrelations exist between the probability for price-up and theprobability for price-down, or between the price change rate and theprice change direction. The conventional technique is not capable ofdescribing the probability density function for the price changedirection as well, and therefore, the probability distribution of theprice change direction for the financial products are disregarded.

Further problems in actual application of the conventional riskevaluation method for financial products relate to insufficientnumerical techniques, such as random number sampling and variancereduction areas for making Monte Carlo calculations. Consequently,undesirable variance inevitably remains in the conventional technique.

Meanwhile, dealers or traders use dealing systems in banks or securitycompanies for purposes of supporting the transactions. The conventionaldealing systems calculate theoretical prices of financial products orits derivatives (hereinafter, referred to as “options”), and simulaterisk evaluation or position change based on accepted theories, such asthe Black-Sholes model or its expanded models. The Black-Sholes modelassumes that the probability distribution of, for example, a stock priceat a feature point of time is the Gaussian (or normal) distribution(Black, F. & M. Sholes, “The Pricing of Options and CorporateLiabilities”, Journal of Political Economy, 81 (May-June 1973), pp.637-59).

However, the conventional dealing systems have many problems listedbelow in item (1) through (8).

(1) A so-called Fat-Tail problem is a serious problem in the financialfield (Alan Greenspan, “Financial derivatives”, Mar. 19, 1999;http://www.federalreserve.gov/boarddocs/speeches/1999/19990319.htm).

In order to calculate theoretical prices for financial products oroptions or to simulate risk evaluation and position change, the usage ofthe Gaussian distribution in the conventional financial engineeringmodels has facilitated evolution of theoretical financial engineeringand implementation to a computer system easy-applicable in businessareas. In other words, non-normality of probability distributions havebeen often observed financial markets, in which big price changesactually occur, or in which transactions are not so active. Once if thisfact would be introduced from the early state, evolution theoreticalfinancial engineering deployment and implementation to a computer systemwould have become much more difficult in actual application. For thisreason, dealers have to transact relying upon their own experiences orintuitions. To carry out such transactions, it is critically importantfor the dealers to accurately grasp the behaviors for volatility of themarket.

(2) There is volatility defined as a historical volatility that iscalculated by the observed fluctuation of price under assettransactions. A general method for obtaining a historical volatility isto calculate the standard deviation of returns for the asset, based onthe observed fluctuation of the closing price. Other known methods forcalculating a historical volatility include so-called “Extreme ValueTheory” for estimating a volatility from daily high prices and lowprices, and a modified Parkinson method for estimating a volatilitywhile taking discontinuity of time in the actual transaction intoaccount.

However, these methods have drawbacks in actual application for somereasons. For example, if transaction is not active enough, there is nocontinuity in the movement for the price of the underlying assets. Incase that the closing prices are used for determining the historicalvolatility, the corresponding exercise time would be altered from theactual one.

Even in the case that the transaction is active, the conventionalmethods are not suitable to estimate the volatility under the fat-tailedregime because these methods assume normality in the risk probabilitydistribution for the market behaviors. To this end, the volatilitycalculated by the conventional methods is used only as a rough guidelinewithin the limited applicability.

(3) Implied volatility (abbreviated as “IV”) is known in the optionmarket, other than the historical volatility mentioned above. Impliedvolatility is volatility calculated back from the option prices observedin the market along with the Black-Sholes equation. Implied volatilityis often used as a factor for calculating the theoretical price of anoption.

However, in a non-active market (for instance, the option transactionmarket for underlying assets in the current Japanese security market),as the number of observed transactions for options is small, the impliedvolatility for the corresponding options cannot be well-defined from theactual market data. For this reason, dealers have to repeatedlymanipulate factors for regular adjusting the volatility parametersthrough their own experiences and expert-judgements in order to reflectthe current market behaviors in their dealing activities.

(4) Implied volatility of an individual stock option can provide somekind of important information to know what is the market reaction to thevolatility of a particular stock for dealers, while the market reactionvaries as transactions go on reflecting market circumstances.

(5) In the option market, different sets of the implied volatility areobtained for multiple options originated from the same originalunderlying asset. This phenomenon is referred to as a “smile effect”. Inthis case, approximated smile curves are drawn in the two-dimensionalphase space with the combinations between a vertical axis showing themagnitude of implied volatility and a horizontal axis showing theexercise prices for the options. Those smile curves are often used tocalculate theoretical prices of the option.

A volatility matrix, which is a data table having a time dimension alongmaturity of the option, in addition to the two-dimensional phase spacementioned above, provide information to obtain the theoretical exerciseprice and to interpolate the volatility value for the regions unobservedin the market behaviors up to maturity, in conjunction with theabove-mentioned item (4).

(6) In fact, in order to obtain reliable smile curves or a volatilitymatrix, sufficient numbers of option prices must be observed in themarket. On the other hand, in some cases that under the transactions inmoderately active market, the option prices observed in the market arescattered in a wide range, it becomes difficult to grasp a comprehensivetrend.

(7) In general, the amount of information observed in the market tradesoff the rationality needed in the basic assumption of the required modelfor estimating theoretical price for the options relating to theunderlying asset price. If numbers of the past records for the optionprices observed in the market are insufficient, a stronger assumption isrequired in a model used to obtain dynamics of an expected probabilitydistribution for the underlying asset price. As one of the advancedmodels having strong assumptions, the stochastic volatility model (SVM)(Hull, John C. & Allan White, “The Pricing of Options on Assets withStochastic Volatilities”, Journal of Finance, 42, June, 1987, pp.281-300), and the GARCH model (T. Bollerslev, “GeneralizedAutoregressive Conditional Heteroskedasticity”, Journal of Econometrics,Vol. 31, 1986, pp. 307-327) are well-known. However, because thesemodels assume normality in the probability distributions, they can notdeal with the Fat-Tail problem sufficiently.

There is a technique expanded from a lattice method and having noassumption of normality (Rubinstein, Mark, “Implied Binomial Trees”,Journal of Finance, 49, July 1994, pp. 771-818). This technique iscapable of forming a flexible probability distribution taking the smileeffect into account. However, this technique requires sufficient numbersof option prices observed in the market in order to determine thedistribution form. Therefore, this technique is not suitably used in anon-active option market.

There is also a famous model named a jump model, which independentlygenerates a stochastic process entirely different from the normaldistribution for a Fat-Tail problem (e.g., R. C. Merton, “Option PricingWhen Underlying Stock Returns Are Discontinuous”, Journal of FinancialEconomics, Vol. 3, March 1976, pp. 125-144). However, the jump model hasan assumption of discontinuous price fluctuation, and therefore, thestochastic volatility model (SVM) naturally becomes a nonlinear problem.For this reason, the risk-neutral measure can not be achievedinvariably, which prevent the option price from being defined uniquely.

(8) In conclusion, no conventional techniques can provide minute andaccurate information in real time for solving the Fat-Tail problem andbeing applicable to a non-active market, although it has been desiredfor dealers and traders to receive significant smile curves or avolatility matrix on their displays in real time in response to theactual market that changes every moment. The conventional technique isincapable of automatically acquiring necessary data required forcomputation in response to requests from the dealers in an interactivemanner, and of automatically selecting the optimum model to analyze themarket deeply and flexibly.

Therefore, an object of the present invention is to introduce aprobability density function with a higher accuracy in comparison withthe normal distribution, and to develop a system capable of correctlyevaluating the price distribution and the risk distribution for afinancial product or its derivatives.

Another object of the present invention is to provide a price and riskevaluation system for a financial product or its derivatives, whichsystem is capable of theoretically solving the above-mentionedheterogeneous or nonlinear problems.

It is still another object of the present invention to provide a priceand risk evaluation system for a financial product or its derivatives,which system introduces a new function of probability density forestimating a price distribution and a risk distribution. Thisprobability density function model can adequately define and describevariables that can not be dealt sufficiently by the conventionaltechnique, and can establish a reliable method.

It is still another object of the present invention to provide a priceand risk evaluation system for a financial product or its derivatives,which system introduces a new function of probability density forestimating a price distribution and a risk distribution of a financialproduct. This function is capable of establishing a sampling method forimproving the efficiency of computation, and allows risk prices to becomputed at a high efficiency.

It is yet another object of the present invention to provide a price andrisk evaluation system for a financial product or its derivatives, whichsystem is applicable to a parallel computing with high efficiency.

It is yet another object of the present invention to provide acomputer-readable recording medium storing a dealing program, whichincludes a Boltzmann model computation developed by the nuclear reactortheories and applied to the financial field, in place of the generaltheories used in the conventional techniques. This program is capable ofdealing with big price changes in the underlying assets (a fat-tailproblem mentioned above), and is applicable to an option market in whichtransactions are not so active. This program allows a computer system todisplay significant theoretical prices and risk parameters on displayterminals of dealers and traders by means of the interactive screeninterfaces.

SUMMARY OF THE INVENTION

The first object of the present invention is to provide a price and riskevaluation system for evaluating a price distribution or a riskdistribution for a financial product or its derivatives withapplications of a new calculation method, the details of which will bedescribed below. A risk calculation method is also provided, which iscapable of treating the fat-tail problems in the financial engineeringfield associated with underlying assets and their derivatives, and ofsubstantially eliminating defects or drawbacks having existed in theprior art.

This system has an initial value setter and an evaluation conditionsetter. The initial value setter receives at least one of the initialvalues of a price, a price change rate, and a price change direction fora financial product or its derivatives that be evaluated. The evaluationcondition setter allows a user to input evaluation conditions includingat least one set of time steps and the number of trials forcalculations. As a significant feature, the system has a Boltzmann modelanalyzer, which receives at least one of the initial values and theevaluation conditions, and repeats simulations of price fluctuationbased on a Boltzmann model using a Monte Carlo method within the rangesof the given calculation conditions. The Boltzmann model analyzer canobtain a price distribution or a risk distribution for the financialproduct or its derivatives in an accurate manner. The risk evaluationsystem also has a velocity/direction distribution setter that suppliesprobability distributions of the price, the price change rate, and theprice-change direction for the financial product or its derivatives tothe Boltzmann model analyzer. This system has a random number generatorin the Boltzmann model analyzer, and an output unit that outputs seriesof analysis results from the Boltzmann model analyzer.

The initial value setter acquires the initial values of the price, theprice change rate, and the price-change direction for the financialproduct or its derivatives from a market database that storesinformation about detailed transaction histories, such as exercises andask-bit data. The initial value setter then supplies the acquiredinitial values to the Boltzmann model analyzer. The velocity/directiondistribution setter receives the past records of a selected financialproduct or its derivatives from the market database, and generates aprobability density function with variables of the price, the pricechange rate, the price change direction, and time. Thevelocity/direction distribution setter then supplies the probabilitydensity function to the Boltzmann model analyzer.

The price and risk evaluation system further has a totalcross-section/stochastic process setter, which supplies information forsetting a sampling time width of the simulation of price fluctuation tothe Boltzmann model analyzer. In this case, the totalcross-section/stochastic process setter acquires a price fluctuationfrequency and a price change rate of the financial product or itsderivatives from the market database storing information about financialproducts or derivative products. The total cross-section/stochasticprocess setter then inputs a ratio of the price fluctuation frequency tothe price change rate into the total cross-section/term of theBoltzmann's equation.

The velocity/direction distribution setter acquires the past records ofa selected financial product or its derivatives from the market databasestoring information. The velocity/direction distribution setter thaninfers a distribution of the price change rate for the financial productor its derivatives using a Sigmoid function and its approximation forms,and supplies the inferred distribution of the price change rate to theBoltzmann model analyzer.

Alternatively, the velocity/direction distribution setter acquires thepast records of a selected financial product or its derivatives from themarket database, and determines a set of the Sigmoid function parametersto estimate the probability distribution of the price change rate usingthe price change rate data stored in the past records. Thevelocity/direction distribution setter then supplies the distribution tothe Boltzmann model analyzer.

In still another alternative, the velocity/direction distribution setteracquires the past records of a financial product or its derivatives fromthe market database, and estimates a probability distribution of theprice change direction for the financial product or its derivativesusing the past records. The velocity/direction distribution setter thensupplies the probability distribution of the price change direction tothe Boltzmann model analyzer.

This velocity/direction distribution setter infers the probabilitydistribution of the price change direction, taking into account acorrelation between the probability for price-up and the probability forprice-down.

In still another alternative, the velocity/direction distribution setteracquires the past records of a financial product or its derivatives fromthe market database, and generates a probability distribution of theprice change direction, taking into account a correlation between thedistribution of the price change rate and the distribution of the pricechange direction for the financial product or its derivatives. Thevelocity/direction distribution setter then supplies the probabilitydistributions to the Boltzmann model analyzer.

In still another alternative, the velocity/direction distribution settergenerates homogeneous probability distributions that are independent ofthe prices, or heterogeneous probability distributions that depend onthe prices, with regard to the probability of a price change rate and aprice change direction distributions. The velocity/directiondistribution setter then supplies these homogeneous or heterogeneousprobability distributions to the Boltzmann model analyzer.

In this system, the Boltzmann model analyzer obtains the pricedistribution or the risk distribution for the financial product or itsderivatives using either a linear Boltzmann model or a non-linearBoltzmann model. In the linear Boltzmann model, the cross-section usedin the Boltzmann's equation is independent of a probability density orflux for the financial product or its derivatives. In the non-linearBoltzmann model, the cross-section for the Boltzmann's equation isdependent on the probability density or the flux for the financialproduct or its derivatives.

Alternatively, the Boltzmann model analyzer obtains the pricedistribution or the risk distribution for the financial product or itsderivatives using a product of a probability density function and aprice change rate per unit time for the financial product or itsderivatives, as flux of the Boltzmann's equation.

The Boltzmann model analyzer evaluates a probability density at anarbitrary time based on the track-length calculated using flux for thefinancial product or its derivatives in order to reduce a variance.

The Boltzmann model analyzer evaluates a price probability in aninfinitesimal price-band or a risk probability in an infinitesimal timeinterval using all of or a part of the price fluctuation data for thefinancial product or the derivatives. Thus, the Boltzmann model analyzerreduces a variance of the price or the risk by applying the pointdetector technique often employed in a neutron transport Monte Carlosimulation.

The Boltzmann model analyzer calculates an adjoint probability densityor an adjoint flux deduced from an adjoint Boltzmann equation for aprice fluctuation of the financial product or the derivatives, andreduces variance by weight-sampling technique using values in proportionto the adjoint probability density or the adjoint flux.

In the price and risk evaluation system for a financial product and itsderivatives, the velocity/direction distribution setter generates avelocity distribution or a direction distribution for a financialproduct or its derivatives, taking into account the inter-correlationamong other multiple financial products or their derivatives. Thegenerated probability distribution functions are supplied to theBoltzmann analyzer.

The Boltzmann analyzer evaluates a price distribution or a riskdistribution for a financial product, and then, applies the Ito'stheorem to calculate the equivalent price distribution or the equivalentrisk distribution for the derivative of that financial product.

Preferably, the Boltzmann analyzer consists of multi-methods forcarrying out simulation of price fluctuations and constructs the editedprobability density function through gathering the simulated pricefluctuations.

According to the present invention, a price or a risk evaluation systemof the first embodiment described above can correctly evaluate theprobability of occurrence of big price changes for a financial product,compared with the conventional technique applying the normaldistribution as for the probability density. Consequently, the pricedistribution or the risk distribution for the financial product or itsderivatives can be evaluated more accurately.

The present system can treat probability densities in heterogeneousproblems in which the probability density function used for evaluationvaries depending on the price, or in nonlinear problems in which theprobability density function itself is nonlinear, without heavilyrelying on experiences or know-how.

The present system also can evaluate probability distributions in moreflexible manners than the conventional methods, especially in thefollowing situations:

non-Gaussian probability distributions, which were not properly takeninto account in the conventional methodology, are treated;

price probability distributions are influenced by the past price itself;

price probability distributions are characterized by theinter-correlation between the probabilities of price-up and price-down;and

probability distributions are correlated with the price change rates andthe price change direction.

The present system requires no time grid for simulation of pricefluctuation, whereas the conventional techniques need time grids for thecalculations. The system can evaluate a probability distribution at anarbitrary point of time within the observed area by introducing the fluxconcept, whereas the conventional technique can treat a probabilitydistribution only at a selected time.

The present system also introduces an idea similar to the point detectorused in the neutron transport calculation by Monte Carlo simulations.The idea of point detector allows the system to automatically detect aroute of causing an event in a target region in an infinitesimalobservation area, in which no price change occurs or no flux passes,based on all of or a part of the price-changing events. Accordingly, thesystem can evaluate an even in an arbitrary infinitesimal observationregion, while reducing the variance.

The present system also reduces the variance generated in the MonteCarlo calculation for a probability density by introducing the idea ofadjoint probability density or adjoint flux into the financialtechnology and selecting weights in proportion to the magnitudes ofadjoint flux in the phase space.

In the situations where inter-correlations are found among theprobability distributions of multiple financial products, the presentsystem can evaluate a price distribution or a risk distribution takingsuch correlations into account. The system includes application of theIto's theorem to evaluate the probability distribution of the pricechange rate for derivatives based on the probability distribution forthe underlying asset.

The present system is realized with parallel computation systems.Consequently, a price distribution or a risk distribution for afinancial product or its derivatives can be obtained in a highlyefficient manner based on the parallel computation.

The present system introduces the Boltzmann model, instead of diffusionmodels having an assumption of the standard normal distribution used inthe conventional systems. Accordingly, the present system can bereplaceable or substitutable for the existing financial-relating systemsfor evaluating risks or analyzing portfolio. This means that thehardware resources and the various types of information required in theexisting system and installed by the conventional methodology can beutilized as they are. As a result, an efficient system for evaluating aprice and a risk probability distribution for a financial product or itsderivatives can be realized.

As the second embodiment of the present invention, a computer-readablerecording medium storing a program of price and risk evaluation isprovided. By installing this program into a computer and causing thecomputer to execute the following operations, a price/risk evaluationsystem can be built up. Namely, an initial value setter of the computersystem inputs at least one of the initial values of a price, a pricechange rate, and a price change direction for a financial product or itsderivatives. An evaluation condition setter of the computer inputsevaluation conditions including at least time steps and the number oftrials. A Boltzmann model analyzer of the computer repeats simulationsof price fluctuation using a Monte Carlo method based on the Boltzmannmodel within the ranges of the evaluation conditions in order to obtaina price distribution or a risk distribution for the financial product orits derivatives. A velocity/direction distribution setter supplies theprobability distributions of the price, the price change rate, and theprice change direction for the financial product or its derivatives tothe Boltzmann model analyzer. A random number generator yields a seriesof random numbers used for a Monte Carlo analysis in the Boltzmann modelanalyzer, and an output unit provides various types of outputs from theanalysis result obtained by the Boltzmann analyzer.

The third embodiment of the present invention covers a computer dealingsystem applying the Boltzmann method and its related calculation tools.The dealing system comprises an implied volatility computation engine, aBoltzmann model computation engine, a conversion filter, and a dealingterminal. The implied volatility computation engine provides an impliedvolatility based on market data. The Boltzmann model computation engineevaluates an option price for a selected option product based on theBoltzmann model using the market data. The conversion filter of thedealing system converts the option price obtained by the Boltzmann modelcomputation engine into a volatility of the Black-Sholes equation. Thedealing terminal provides displaying the volatility of the Black-Sholesequation in comparison with the implied volatility calculated from themarket data, or displaying the option price calculated by the Boltzmannmodel computation engine in comparison with an option price in market.

The present dealing system defines a unique and risk-neutral probabilitymeasure by applying the Boltzmann model used in the financialengineering field to option price evaluation, because the system cantreat Leptokurcity and Fat-tail problems for a price or a riskprobability distribution appropriately in a linear equation form.Consequently, the system can evaluate option prices with therisk-neutral and unique manner taking into account the Leptokurcity andFat-tail of the price-change distribution. Applying the Boltzmann modelto the option price evaluation of a selected option product allows thesystem to grasp the comprehensive tendency of a volatility matrix fromthe past transaction records varying in a wide range.

The present dealing system covers dealing of a stock price index optionor an individual stock option as the option product. Consequently, thecomprehensive tendency of the volatility matrix of the individual stockoption can be obtained.

As described above, the present dealing system provides whole trends forvolatility matrices for the individual stock option whose dealings areless active, by checking the consistency of the daily earning rates withthe corresponding underlying assets. This can be achieved because theBoltzmann model is capable of pricing the options through determining aset of Boltzmann parameters as to reproducing the daily earningprobability reflecting the market data for the underlying assets.

The Boltzmann model computation engine of the present dealing system hasa calculation unit that calculates an option price consistent withhistorical information. Accordingly, a well-adjusted option price can beprovided to the user via the dealing terminal.

In the present system, the Boltzmann model computation engine also has aconverter that converts option prices, which were sets of exerciseprices for the discrete months of the delivery, into sets of equivalentvolatility from the Black-Sholes equation. The equivalent option pricesand the risk parameters are obtained through interpolation of theBlack-Sholes equation, and are displayed on the dealing terminal of theuser.

The Boltzmann model computation engine in the present system has a tablegenerator that generates a table of a probability density functionevaluated by the Boltzmann model, and calculates an option price fromthe sum of inner product of vectors (i.e., Riemann sum). Thisarrangement increases the operation speed, and realizes a highlyresponsive dealing system.

As the fourth embodiment of the present invention, a dealing systemcomprising a dealing terminal, a theoretical option price and parametercomputation engine, an interpolation unit, and an interface thatreceives market data. The dealing terminal functions as a graphics userinterface. The theoretical option price/parameter computation engine isconfigured to switch between a rough computation and a detailedcomputation of a theoretical option price and parameters. The roughcomputation provides the theoretical option price and the parameters foreach exercise price and for each delivery month set in the normalmarket-state. The detailed computation provides more detailedinformation including the theoretical option prices and the parametersfor those exercise prices and the specified delivery months by usersthat are not set in the market, but designated by a user. The priceinterpolation unit covers pricing with the Boltzmann model up toarbitrary maturities specified by users. This dealing system providesthe rough computation results to the dealing terminal as a marketactivity at a high speed in the ordinary state. The dealing systemdisplays the detailed computation results on the dealing terminal for aspecific price band specified by the user.

In the present dealing system, quick market display is made for therough computation results in the normal state, rather than displayingthe detailed computation results. On the other hand, the detailedcomputation results are displayed to timely inform the user of anychanges in the market activity, by calculating the detailed theoreticalprice and parameters based on the Boltzmann model within the associatedprice range in response to the user's instruction.

As an alternative, the dealing system comprises a dealing terminal, arough computation engine, a multi-term Boltzmann engine, aninterpolation unit, and an interface. The dealing terminal functions asa graphics user interface. The rough computation engine computes atheoretical option price and parameters for an exercise value of eachdelivery month set in the market. The multi-term Boltzmann enginecomputes theoretical option prices and parameters at arbitrary termsbased on the Boltzmann model. The dealing system normally causes thedealing terminal to display the market activity based on the roughcomputation result, and causes the display terminal to show themulti-term volatility in response to the user's instruction. The roughcomputation results are made faster than the detailed computationresults. Because the present system can evaluate the term structure ofvolatility that does not come up in the market data until the end oftime concerned, the developing efficiency of a structured bond or anexotic option can be improved.

As the fifth embodiment, a dealing system, which comprises a dealingterminal having a graphics user interface, a rough computation engine, adetailed computation engine, an interpolation unit, a position setter,an automatic transaction order unit, and an interface for receivingmarket data, is provided. The rough computation engine computes atheoretical price and an index for each exercise price and for eachdelivery month set in the market. The detailed computation enginecomputes detailed information including theoretical prices andparameters for exercise prices and delivery months that are not set inthe market. This dealing system outputs an automatic order signal when astock index option price or an individual stock option price reaches apredetermined automatic ordering price band. This system allows the userto visually confirm the appropriate standard with an advanced model, toset a position, and to timely order in an automatic manner.

The present dealing system with the graphical user interface (i.e., thedealing terminal) facilitates a fading processor as an alternative. Inthis case, the dealing system causes the dealing terminal to display ananimated behavior with a fading style for a term structure of avolatility that has been converted from an option price at the money(ATM) obtained from the Boltzmann model. This arrangement can preventmispricing due to overreactions to the market behaviors.

The present dealing system installs a risk limit setter. In this case,the dealing system alerts the user when the price in the market entersthe warning area specified by the user. In other words, the systemallows the user to set a risk limit, and enables the dealers concernedto conduct risk management appropriately.

The present dealing system installs an alternative-position selector, inaddition to the risk limit setter unit. In this case, the dealing systemalerts the user, and simultaneously causes the dealing terminal todisplay an alternative position. Since the alternative position isautomatically selected, the system can prevent the user from suffering aloss due to overreactions to the fluctuations in the market price.

As the sixth embodiment of the present invention, a computer-readablerecording medium storing a dealing program is provided. By installingthis program in a computer and causing the computer to execute thefollowing operations, a dealing system can be built up. Such a dealingsystem computes an implied volatility and an option price of a selectedasset based on the Boltzmann model using the market data. The systemthen converts the option price obtained from the Boltzmann model into anequivalent volatility from the Black-Sholes equation. Finally, thesystem displays the equivalent volatility from the Black-Sholes equationin comparison with the implied volatility calculated from the marketdata, or to display the option price based on the Boltzmann model incomparison with an option price in market.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 represents a block diagram showing the structure and theoperation flow of a price and risk evaluation system for financialproducts according to the present invention;

FIG. 2 illustrates an operation flow of the Boltzmann model analysisunit according to the present invention;

FIG. 3 illustrates another operation flow of the Boltzmann modelanalysis unit which uses a price change frequency; according to thepresent invention;

FIG. 4 schematically illustrates the simulation results of the operationflow of FIG. 2 in the observation areas;

FIG. 5 schematically illustrates the simulation results of the operationflow of FIG. 3 in the observation area;

FIG. 6 is a graph showing a probability distribution simulating adiffusion model using the Boltzmann model according to the presentinvention;

FIG. 7 is a graph showing price changes simulating the diffusion modelusing the Boltzmann model according to the present invention;

FIG. 8 is a graph showing required spectra with respect to the pricechange rate v of stock prices;

FIG. 9 is a graph showing the dependency of the spectra on the incidentvelocity v′;

FIG. 10 is a graph showing the dependency of the price-up component(positive changes) of the spectra on the incident velocity v′;

FIG. 11 is a graph showing the dependency of the price-down component(negative changes) of the spectra on the incident velocity v′;

FIG. 12 is a graph showing an application of evaporation spectra;

FIG. 13 is a graph showing the empirical equation of the velocity termof the differential cross-section;

FIG. 14 illustrates the averages of the continued price-up probabilityand the continued price-down probability of every 5 days;

FIG. 15 is a graph of the simulation result of price fluctuation using aBoltzmann model;

FIG. 16 shows the simulation result of price fluctuation using theBoltzmann model with a detailed view;

FIG. 17 illustrates the distribution of the stock price after twohundred days using the Boltzmann model;

FIG. 18 is a graph of stock price distributions of every twenty daysusing the Boltzmann model;

FIG. 19 illustrates the configuration of parallel processing of theprice and risk evaluation system for a financial product or itsderivatives according to the present invention;

FIG. 20 is a graph showing the price fluctuation C1 of the underlyingassets expected by the geometric Brownian model, in comparison with theclosing-price fluctuation (daily earning rate) C2 of a typical stockprice;

FIG. 21 is a graph of fluctuation of the daily earning rate C3 of thestock price average of the Nikkei 225 stock average;

FIG. 22 is a graph showing the implied volatility of the put option ofthe price index of stocks of the Nikkei 225 stock average, together witha smile curve;

FIG. 23 is a graph showing the probability density of an actual dailyearning rate, together with the normal distribution presumed byBlack-Sholes equation;

FIG. 24 is a graph showing the price change probability estimated by theBoltzmann model, together with the logarithmic normal distribution usedin Black-Sholes equation;

FIG. 25 is a graph of temperature T as a function of daily earning ratev′ of the previous day;

FIG. 26 is a graph of simulated probability density as a function ofdaily earning rate for the price valuation using the Boltzmann model;

FIG. 27 is a graph showing the implied volatility of the Boltzmannmodel, in comparison with a jump model;

FIG. 28 is a block diagram for the dealing system according to thepresent invention;

FIG. 29 is a block diagram showing the operation flow and the structureof the Boltzmann model computation engine used in the dealing systemshown in FIG. 28;

FIG. 30 is a flowchart of theoretical computation carried out by thedealing system shown in FIG. 28;

FIG. 31 is a graph of the temperatures T of various listed stocks ofTokyo stock exchange obtained by the dealing system of the invention;

FIG. 32 shows examples of evaluation for call option pricing ofindividual stock options, which is expressed by ratio of call optionprice to underlying assets as a function of ratio of exercise price tounderlying assets;

FIG. 33 shows examples of evaluation of put option pricing of individualstock options, which is expressed by ratio of call option price tounderlying assets as a function of ratio of exercise price to underlyingassets;

FIG. 34 is a graph indicating the relationship between the ratio ofexercise price of a call option to underlying assets and the impliedvolatility, which is obtained by the dealing system of the invention;

FIG. 35 is a graph indicating the relationship between the ratio ofexercise price of a put option to underlying assets and the impliedvolatility, which is obtained by the dealing system of the invention;

FIG. 36 shows the operation flow of theoretical computation flow carriedout by the dealing system according to the invention;

FIG. 37 illustrates a sub-screen displayed on a terminal of the dealingterminal of system, which displays the detailed track of stock index inthe continuous session;

FIG. 38 illustrates a sub-screen displayed on a terminal of the dealingsystem, which displays a table of the implied volatility, the marketprices for each delivery month, and each exercise price of stock indexoption, together with the underlying assets;

FIG. 39 illustrates sub-screens of the dealing system, which displays,in graphs, information contained in the table, such as the impliedvolatility, the market prices of stock index options for each exerciseprice and each delivery month, using the stock indices displayed on theterminal as an underlying asset price;

FIG. 40 illustrates an operation flow of the detailed price evaluationcarried out by the dealing system;

FIG. 41 illustrates switching of graphs displayed on a terminal of thedealing system during the process of detailed price evaluation shown inFIG. 40;

FIG. 42 illustrates switching of tables displayed on a terminal of thedealing system during the process of detailed price evaluation;

FIG. 43 is a graph representing the theoretical computation carried outby the Boltzmann model computation engine of the dealing system;

FIG. 44 illustrates an operation flow of the evaluation process of anarbitrary multi-term volatility carried out by the dealing system;

FIG. 45 illustrates a period setting screen displayed on a terminal ofthe dealing system in the evaluation process of the arbitrary multi-termvolatility shown in FIG. 44;

FIG. 46 illustrates a table with market prices and implied volatilityvalues for the arbitrary multi-term delivery in the dealing system;

FIG. 47 is a graph showing an example of estimation result for thearbitrary multi-term volatility, which is displayed on the terminal ofthe dealing system as a result of the evaluation process;

FIG. 48 illustrates an operation flow in a modified process for thedetailed price evaluation with a fading operation carried out by thedealing system, which includes a fading operation;

FIG. 49 illustrates an operation flow showing the detailed steps for thefading operation;

FIG. 50 is a table of the implied volatility and the option prices, inwhich a virtual implied volatility for a real-time ATM is inserted inthe exercise price band set in the marked, used in the detailed priceevaluation process carried out by the dealing system;

FIG. 51 is a graph of implied volatility for arbitrary multi-terms,which includes the implied volatility for a virtual real-time ATMinserted in the exercise price band set in the market, displayed in thedetailed price evaluation process carried out by the dealing system;

FIG. 52 is a process diagram for explaining the fading operation;

FIG. 53 illustrates an operation flow in the arbitrary multi-termvolatility evaluation process using the fading function;

FIG. 54 illustrates an operation flow for an automatic ordering processcarried by the dealing system, in which a deal sets a desired positionand orders timely;

FIG. 55 is a graph showing the relationship between the output from thecomputation engine and the implied volatility, which is used for settinga position;

FIG. 56 illustrates a input screen displayed on a terminal of thedealing system with the operation for position setting;

FIG. 57 illustrates an operation flow in the first part of the processfor displaying the simulating animation for the term structure analysisof the ATM implied volatility in a fading manner in the dealing system;

FIG. 58 illustrates an operation flow in the second part of the processfor displaying the simulating animation for the term structure analysisof the ATM implied volatility in a fading manner;

FIG. 59 illustrates an operation flow in the third part of the processfor displaying the simulating animation for the term structure analysisof the ATM implied volatility in a fading manner;

FIG. 60 illustrates an operation flow for the detailed steps of theinterpolation A shown in step S30 of FIG. 58;

FIG. 61 illustrates an operation flow for the detailed steps of theinterpolation B show in step S30′ of FIG. 58;

FIG. 62 is a graph showing the approach for setting the automaticordering process;

FIG. 63 illustrates an operation flow for setting an automatic warningfunction with market data in the dealing system;

FIG. 64 is a graph showing the approach for setting the automaticwarning process;

FIG. 65 illustrates an operation flow for automatically computing andselecting an alternative position, which is carried out together withthe automatic warning process in the dealing system; and

FIG. 66 is a graph showing the approach for setting the automaticwarning process, together with selection of an alternative position inthe dealing system.

BEST MODE FOR CARRYING OUT THE INVENTION

The preferred embodiment of a price and risk evaluation system for afinancial product or its derivatives will now be explained withreference to the drawings. FIG. 1 illustrates both the structure and theoperation flow of the price and risk evaluation system for financialproducts.

The price and risk evaluation system 1 includes a portfolio setup unit2, an initial value setup unit 3 for inputting a price, a price changerate, and a price change direction, and an evaluation condition setupunit 4, a Boltzmann model analysis unit 5, an output unit 6, an totalcross-section/stochastic process input unit 7, a velocitydistribution/direction distribution setup unit 8, a random numbergenerator 9, a VaR-evaluation unit 10, and a market database 11.

The Boltzmann model analysis unit 5 includes an initialization unit 12,an initial value setting unit 13, a sampling unit 14, a pricefluctuation simulation unit 15 for simulating price fluctuation based onthe Boltzmann model, a probability density calculating unit 16, aone-trial-completion detector 17, an all-trial-completion detector 18,and a probability density editor 19.

The rectangle defined by the long dashed line indicates the evaluationsystem 1 of the embodiment. The market database 11 and theVaR-evaluation unit 10 are positioned across the long dashed, whichmeans that these elements can be connected as external units to theevaluation system 1 via data communication.

The long dashed line does not intend to define a physical boarder ofcomputer. For example, the elements included in the evaluation system 1can be appropriately divided if the system is designed to carry outdispersed operations, such as a client-server system.

The portfolio input unit 2 receives portfolio, and outputs a financialproduct or its derivatives for evaluation.

In an ordinary money management, the assets to be invested are allocatedto multiple financial products or their derivatives in order to reduce arisk, and to carry out the most advantageous money management as awhole. A set of such multiple financial products or the derivatives, orthe combination of these, is named portfolio. The portfolio input unit 2extracts the financial product or its derivatives that is to beevaluated among from the portfolio, and outputs the extracted product.

Preferably, the portfolio input unit 2 has a portfolio table or databaseinside it, and allows a user to input the ID code of a desiredportfolio. The portfolio setup unit 3 then exhibits the configuration ofthe portfolio, and allows the user to select a financial product or itsderivatives concerned.

The portfolio input unit 2 is not an essential element of the presentinvention, and therefore, it may be omitted if the data required forevaluating the selected financial product or its derivative is known.

The initial value setup unit 3 supplies at least one of the initialvalues of the price, the price change rate, and the price changedirection for the financial product or its derivatives to be evaluated,to the Boltzmann model analysis unit 5.

The initial values of the price, the price change rate, and the pricechange direction for the financial product or its derivatives areobtained from the past record. Preferably, the initial value setup unit3 receives the financial product or the derivatives from the portfoliosetup unit 2, and retrieves information as to the financial product orits derivatives from the market database 11. The initial value setupunit 3 then acquires the initial values of the price, the price changerate, and the price change direction for this financial product or itsderivatives from the past records contained in the retrievedinformation. The acquired initial values are supplied to the Boltzmannmodel analysis unit 5. The initial-value input unit 3 is an essentialelement of the price/risk evaluation system 1.

The evaluation condition setup unit 4 supplies the evaluation conditionsto the Boltzmann model analysis unit 5. The evaluation conditionsinclude, for example, the number of trials, the time zone, and the priceband for evaluation, which are required for analysis by the Boltzmannmodel analysis unit 5. The evaluation condition setup unit 4 is anessential element of the evaluation system 1, and supplies anyevaluation conditions required by the Boltzmann model analysis unit 5.

The Boltzmann model analysis unit 5 is the center of the price and riskevaluation system 1. The Boltzmann model analysis unit 5 receives theinitial values and the evaluation conditions from the initial valuesetup unit 3 and the evaluation condition setup unit 4, respectively,and repeats price fluctuation simulations for the selected financialproduct or its derivatives. The price fluctuation simulation is carriedout based on the Boltzmann model within the range of evaluationcondition, using the Monte Carlo method. The Monte Carlo method is anumerical analysis method for obtaining a rigorous solution of aBoltzmann equation.

The initialization unit 12 of the Boltzmann model analysis unit 5initializes the values of the price, the price change rate, and theprice change direction for the financial product and its derivatives tostarting evaluation.

The initial value setting unit 13 of the Boltzmann model analysis unit 5sets up the initial values of the price, the price change rate, and theprice change direction of the financial product based on the inputs fromthe initial value setup unit 3.

The sampling unit 14 of the Boltzmann model analysis unit 5 determines asampling width of the price fluctuation simulation. As the feature ofthe present invention, the sampling unit 14 can set a probability ofprice change per unit time based on the input from the totalcross-section/stochastic process setup unit 7. This arrangement allowssetting of a time grid for simulation to be omitted. Setting of a timegrid was required for simulation in the conventional system, but is adifficult technique. This will be described in more detail below.

The price fluctuation simulation unit 15 of the Boltzmann model analysisunit 5 simulates the price distribution from the last price based on theprobability distributions of the price change rate and the price changedirection, using the Monte Carlo method.

The price fluctuation simulation unit 15 receives the price change rateand the price change direction for the financial product from the pricechange rate distribution and the price change direction distributionsetup unit 8, in order to simulate a price distribution based on aBoltzmann model. The price change rate and the price change directioncorrespond to the velocity/direction distributions in the Boltzmannequation.

The price fluctuation simulation unit 15 also receives a series ofrandom numbers generated by the random number generator 9, and computesthe solution of the Boltzmann equation by the Monte Carlo method.

The probability density calculating unit 16 of the Boltzmann modelanalysis unit 5 integrates the price distribution simulated by the pricefluctuation simulation unit 15 to obtain the probability density.

The one-trial completion detector 17 of the Boltzmann model analysisunit 5 determines whether a trial has been completed. In this context,“one trial” means a series of simulation for price fluctuations in thetime period from the beginning to the end of the simulation process. Theone-trial completion detector 17 can detect the completion of one trialby comparing the current point of time in the process simulation. Thecondition for completion of one trial is supplied from the evaluationcondition setup unit 4.

If one trial has not been completed yet for a series of simulation, theprocess returns to the sampling unit 14 to calculate the price and theprobability density for the subsequent time regime using the previousprice, and distributions of the price change rate and the price changedirection.

The all-trial completion detector 18 of the Boltzmann model analysisunit 5 determines whether or not the current number of trials hasreached the selected number of trials given by the evaluation conditionsetup unit 4. The maximum trial number is supplied to the all-trialcompletion detector 18 from the evaluation condition setup unit 4.

The probability density editor 19 of the Boltzmann model analysis unit 5collects the probability densities obtained from the entire trials andedits the probability density of the price fluctuation for the financialproduct or its derivatives. If the Boltzmann model analysis unit 5 hasmultiple simulators for obtaining price fluctuations in different waysin addition to the Boltzmann models, as will be explained below, theprobability density editor 19 collects the probability densities fromeach simulator.

The foregoing is the explanation for the outline of the major componentsof the Boltzmann analysis unit 5. The detailed operation flow of theBoltzmann model analysis unit 5 will be explained later again.

The output unit 6 presents the calculation results obtained from thesystem, such as the price distribution, the risk distribution, and theintegrated risk index for the selected financial products. The outputunit 6 is any known output unit as long as it can output the operationresults with some forms. For example, the output unit 6 can be a printeras a hard copy, a display monitor as an image, or any communicationmeans as an external data file. The output unit 6 can also outputintermediate results from the Boltzmann model analysis unit 5, such asthe price fluctuation simulation results or the probability densitydistribution at the given trial. The output unit 6 may physicallyincludes multiple output means.

The total cross-section/stochastic process setup unit 7 supplies a setof sampling time intervals through setting fluctuation probabilities (orfrequencies) per unit time at the sampling unit 14, as has beenexplained in conjunction with the sampling unit 14 of the Boltzmannanalysis unit 5.

The total cross-section/stochastic process setup unit 7 acquires thefrequencies of price change and the price change rates for the selectedfinancial product or its derivatives from the market database 11, andsupplies total cross sections, which are defined as the ratios of thefrequency of price change to the price change rate, to the Boltzmannequation.

The total cross-section of the Boltzmann equation corresponds to thefrequency of price change for the financial product or its derivatives,which will be described in more detail below. If a sampling time grid isused instead of the frequency of price change, as in the conventionalsystem, then the total cross-section/stochastic process setup unit 7 canbe omitted.

The velocity distribution/direction distribution setup unit 8 suppliesthe distributions of the price change rate and the price changedirection to the simulator 15, as has been explained.

The velocity distribution/direction distribution setup unit 8 acquiresthe past records for the financial product from the market database 11,and obtains the distributions of the price change rate and the pricechange direction from the past data. The distributions are then suppliedto the simulator 15.

Preferably, the velocity distribution/direction distribution setup unit8 has a numerical analysis function. With this function, thedistribution setup unit 8 is capable of estimating the distribution ofprice change rate from the past data using a sigmoid function and itsapproximate function. Alternatively, the distribution setup unit 8 candetermine a set of parameters for the sigmoid function for thedistribution of the price change rate using the price change rate of theprevious day. The distribution setup unit 8 also estimates thedistribution of the price change direction taking the correlationbetween the probability of price-up and the probability of price-downinto account, or alternatively, taking the correlation between the pricechange rate and the price change direction into account. Thedistribution input unit 8 is also capable of generating a probabilitydistribution corresponding to the price, if the distributions of theprice change rate and of the price change direction are dependent on theprice.

The random number generator 9 produces a random number used in theBoltzmann model analysis unit 5 to simulate a price fluctuation usingthe Monte Carlo method. The generated random number is supplied to theprice fluctuation simulation unit 15, as has been described above.Application of this random number will be explained below.

The velocity distribution/direction distribution setup unit 8 and therandom number generator 9 are essential elements for the Boltzmannmodel.

The VaR-evaluation unit 10 calculates a risk or a risk distribution fromthe price distribution for a selected financial product or itsderivatives.

Several devices for calculating a risk or a risk distribution for afinancial product or its derivatives from the price distribution for thefinancial product have been conventionally realized as VaR-evaluationdevice. That is, the provability density computed by the Boltzmann modelaccording to the present invention can be transferred to anyconventional VaR-evaluation unit 10, which then provides a pricedistribution, a risk distribution, and a risk value integrated as anindex for the outputs.

The VaR-evaluation unit 10 is not essential to the present inventionbecause any conventional devices may be used as the VaR-evaluation unit10.

The market database 11 stores information about financial products andtheir derivatives. In the specification, the term “database” includesdata itself systematically stored in the database, data search engines,and hardware capable of storing the data.

Although the market database 11 is essential to the present invention,any existing external database may be used. To this end, a system usingan existing database together with the other essential elements of thepresent invention falls into the scope of the present invention.

The foregoing is the configuration of the price and risk evaluationsystem 1 of an embodiment of the present invention. Preferably, theelements conducting data processing may be included in the CPU of acomputer that activates installed programs and controls the respectivetasks. In this case, different processing means may be included in thesame hardware with parallel ways. The input units among theabove-described elements may be an ordinary keyboard or pointing device.If data is acquired from other data files via data communication, thedata communication means itself becomes the input unit.

Hereafter, the theoretical backgrounds are described to clarify thereason why the Boltzmann model introduced into the present invention toconduct risk analysis for financial products can correctly estimate theprice change probability distributions for the financial products withhigher freedom than the conventional financial models.

Basically, the present invention needs to input several parameters, forthe Boltzmann equation, which include an initial price, a distributionof price change rate, a distribution of price change direction, and timedomain concerned for each financial product. With these parameters, theBoltzmann equation is solved using the Monte Carlo method, and the priceand risk distributions obtained at a specific time domain as solutionsof the equation.

First, the time-dependent behaviors of price for a selected financialproduct or its derivatives are described by a Boltzmann equation. Thepresent invention applies a neutron transport Boltzmann equation, whichis generally used to design a nuclear reactor as the establishedmethodology in the nuclear industry.

Neutron transport Boltzmann equation is an equation for describing amacroscopic behavior of neutrons. A model for explaining a phenomenonbased on a Boltzmann equation is called the Boltzmann model.

In a neutron-relating Boltzmann model, the position of a neutron isdefined by the seven-dimensional vectors r, vΩ, and t. Here, r=(x, y, z)denotes a vector in the position space of the Cartesian coordinates,vΩ=(vΩ_(x), vΩ_(y), vΩ_(z)) is vector in the velocity space, and tdenotes time. A set consisting of the seven-dimensional vectors iscalled a phase space.

A general formula of the neutron transport Boltzmann equation is asfollows. $\begin{matrix}{{- \frac{\partial{\phi( {r,v,{\Omega;t}} )}}{v{\partial t}}} = {{\Omega \cdot {\nabla{\phi( {r,v,{\Omega;t}} )}}} + {{\Sigma_{t}( {r,v} )}{\phi( {r,v,{\Omega;t}} )}} - {\int{{\mathbb{d}v^{\prime}}{\mathbb{d}\Omega^{\prime}}{\Sigma_{s}( {r,v^{\prime}, \Omega^{\prime}arrow v ,\Omega} )}{\phi( {r,v^{\prime},{\Omega^{\prime};t}} )}}} - {S( {r,v,{\Omega;t}} )}}} & (6)\end{matrix}$

In the equation (6), the important quantities are Σ_(t)(r, v) andΣ_(s)(r, v′, Ω′→v, Ω), which are a microscopic total cross-section and adouble differential cross-section, respectively. These quantitiesindicate probabilities of neutron collisions and scattering per unitlength.

Microscopic cross-section is a product of an atomic number density (unitis cm⁻³) and a microscopic cross-section in the reactor physics. Themicroscopic cross-section is determined by the nuclide (for example,uranium, oxygen, hydrogen, and so on) existing in a nuclear reactor. Onthe other hand, the microscopic cross-section is an effectivecross-section of a nucleus (unit is cm²) giving the collisionprobability between one nucleus and one neutron. The name“cross-section” is derived from the unit that expresses an area ofnucleus.

Since a concept similar to the atomic number density has not been foundin the financial engineering field, the microscopic cross-section andthe microscopic cross-section can not be distinguished. Thereupon, ifthe neutron transport Boltzmann equation is applied to finance, themicroscopic cross-section and the microscopic cross-section areconsolidated in a single concept for cross-section. In physics, thedouble differential cross-section corresponds to the velocity and theangle distributions of a neutron emitted from the nuclear reactions.

It is impossible to solve an arbitrary Boltzmann equation analyticallywith a deterministic way, because the cross-section changescomplicatedly in the phase space. On the contrary, it is known that theMonte Carlo method is capable of solving the Boltzmann equation withoutany approximation.

To explain the means for solving the problem, equation (6) issimplified, and a one-dimensional and homogeneous problem with nointernal neutron source will be discussed. In the neutron transporttheory under a one-dimensional problem, a homogeneous distribution isassumed the y and z directions. Accordingly, the vector Ω is athree-dimensional vector in the one-dimension problem. In order todiscuss the one-dimensional and homogeneous problem applied to thefinancial field, the Brownian motion of a price for a single stock willbe described hereafter. The direction is denoted by μ indicating “rise”and “fall” only, and the other directions are not defined.

Equation 6 is written as equation (7). $\begin{matrix}{{- \frac{\partial{\phi( {x,v,{\mu;t}} )}}{v{\partial t}}} = {{\mu\frac{\partial{\phi( {x,v,{\mu;t}} )}}{\partial x}} + {{\Sigma_{t}(v)}{\phi( {x,v,{\mu;t}} )}} - {\int{{\mathbb{d}v^{\prime}}{\mathbb{d}\mu^{\prime}}{\Sigma_{s}( {v^{\prime}, \mu^{\prime}arrow v ,\mu} )}{\phi( {x,v^{\prime},{\mu^{\prime};t}} )}}}}} & (7)\end{matrix}$

If the velocity v of equation (7) can be integrated into therepresentative velocity u, and if the angle distribution is isotropic,then equation (7) can be deduced to equation (8). $\begin{matrix}{\frac{\partial{\phi( {x;t} )}}{v{\partial t}} = {D\quad{\frac{\partial^{2}{\phi( {x;t} )}}{\partial x^{2}}.}}} & (8)\end{matrix}$

The Diffusion Coefficient Becomes $\begin{matrix}{D = \frac{1}{f\quad{\Sigma_{t}(u)}}} & (9)\end{matrix}$here, f is the degree of freedom of the system. In this one-dimensionaland homogeneous problem without an internal neutron source, f takes avalue of “1” ideally, because the other directions are undefined.

The flux expression is very convenient for a neutron transport problem.The flux expression gives many advantages to the Monte Carlo simulation.Neutron transport Monte Carlo simulation is characterized by manyeffective variance-reduction techniques. These techniques can beintroduced using flux expression. However, describing a financial MonteCarlo with the flux expression is likely to cause confusion, andtherefore, the conventional density expression will be used temporarily.

A neutron density function p (x, v, μ; t) is given by the solution ofBoltzmann equation (10). $\begin{matrix}{{- \frac{\partial{p( {x,v,{\mu;t}} )}}{\partial t}} = {{v\quad\mu\frac{\partial{p( {x,v,{\mu;t}} )}}{\partial x}} + {{\Sigma_{t}(v)}{{vp}( {x,v,{\mu;t}} )}} - {\int{{\mathbb{d}v^{\prime}}{\mathbb{d}\mu^{\prime}}{\Sigma_{s}( {v^{\prime}, \mu^{\prime}arrow v ,\mu} )}v^{\prime}{p( {x,v^{\prime},{\mu^{\prime};t}} )}}}}} & (10)\end{matrix}$

Equation 8, which is a neutron diffusion equation, is also rewritten as$\begin{matrix}{\frac{\partial{P( {x;t} )}}{\partial t} = {D^{\prime}{\frac{\partial^{2}{P( {x;t} )}}{\partial x^{2}}.}}} & (11)\end{matrix}$

The density function P (x; t) is an integration of p (x, v, μ; t) atvelocity v and angle μ. The diffusion constant in the density expressionbecomes $\begin{matrix}{D = \frac{u^{2}}{f\quad\lambda_{t}}} & (12)\end{matrix}$here, λ_(t) is a frequency of collision between a neutron and a medium,and is expressed asλ_(t) =uΣ _(t)(u).  (13)

If the concept of the diffusion constant D′ and the concept of thevolatility are the same, then equation 12 stands, and the volatility σbecomes $\begin{matrix}{\sigma = {\sqrt{\frac{2u^{2}}{\lambda_{t}}}.}} & (14)\end{matrix}$

Equations 13 and 14 show that the neutron velocity and the totalcross-section are connected to volatility. A typical volatility σ takesa value of square root of 0.1 per year [year⁻¹]. If the price changesonce a day, the collision frequency λ_(t) becomes 365/year. Here, u isevaluated as being 0.0117/day, which is substantially equal to theaverage of the stock-price change rate per day.

A numerical example of the total cross section is indicated by equation(15). $\begin{matrix}\begin{matrix}{{\Sigma_{t}(u)} = \frac{2u}{\sigma^{2}}} \\{= \frac{2 \times {0.0117\quad\lbrack {1\text{/}{day}} \rbrack} \times {365\lbrack {{days}\text{/}{year}} \rbrack}}{0.1\lbrack {1\text{/}{year}} \rbrack}} \\{= 85.4}\end{matrix} & (15)\end{matrix}$

The total cross-section (Σ_(t)) is in inverse proportion to square ofvolatility, as shown in equation (15). This relationship guarantees theequivalence between the volatility in the financial technology and thetotal cross section in the Boltzmann model.

Based on this relationship, the Boltzmann equation can provide a basisfor determining a price or a risk for a financial product or itsderivatives by relating the variables in the neutron transport Boltzmannequation, such as position x, volocity v, angle μ, and time t, with theprice x, the price change rate v per unit time, the price changedirection μ, and the transient time t for a financial product or itsderivatives, respectively.

No concept relating to the concept of neutron double differential crosssection has been defined yet in the financial technology. However, inorder to apply the Boltzmann model to the financial technology, thedouble differential cross section should be defined. In this embodiment,an evaluation technique for a neutron cross section is applied directlyto determine the double differential cross section of a stock price.

The cross section can be evaluated from the experimental data andtheoretical computation due to nuclear physics for the neutron transportproblem. At present, there is no established theory for evaluating thecross section for a stock price, and accordingly, the doubledifferential cross section is determined from the market data, that is,for example, stock prices announced on newspapers, internets, and so on.

In the evaluation, the distribution of the price change rate v per unittime is estimated from the past records for stock prices using a sigmoidfunction and the approximation form.

In the Boltzmann model, the velocity distribution term of the doubledifferential cross section must be determined by defining a sigmoidfunction of the price change rate v using the price change rate v′corresponding to the daily return for the previous day.

Furthermore, in solving the Boltzmann equation, the directiondistribution term of the double differential cross section in Boltzmannequation can be determined as the direction change probability for afinancial product or its derivatives, from the past data for thefinancial product.

If there is a correlation between the price-up (or upspring) probabilityand the price-down (or downfall) probability along the price changedirection for the financial product, this correlation can be taken intoaccount for determining the direction distribution term of the doubledifferential cross section in the Boltzmann equation. Taking thecorrelation into account allows a probability drift to be evaluated indetail.

There are two types of price distributions, namely, a symmetricaldistribution with respect to the expected value, and an asymmetricdistribution with respect to the expected value. A symmetricaldistribution can be obtained by separating variables into the velocitydistribution term and the direction distribution term of the doubledifferential cross section. An asymmetrical distribution can be obtainedby taking the correlation between the distribution of the price changerate and the probability of the direction change into consideration.

Under the Boltzmann model, there are two types of problems, namely, ahomogeneous problem, in which the cross section is independent of aprice, and a heterogeneous problem, in which the cross-section isdependent on the price. The homogeneous problem treats the distributionsof the price change rate and the price change as constant concerning theprice x for the financial product. On the contrary, the heterogeneousproblem treats the distributions of the price change rate and the pricechange direction as variable in response to the price x for thefinancial product.

In this manner, the Boltzmann model is capable of comprehensivelydescribing both cases in which the volatility is dependent andindependent of the price, by applying the Boltzmann equation to ahomogeneous problem and a heterogeneous problem. In contrast, it wasdifficult for the conventional technique to treat such a realisticsituation with the volatility dependent on the price.

The Boltzmann model is also capable of treating both a linear problem,in which the cross section is independent on the probability density orthe flux for the financial product, and a nonlinear problem, in whichthe cross section is dependent on the price with a consistent manner.

Again, the Boltzmann model can comprehensively describe both cases, inwhich the volatility is dependent on the price as non-linear problem andindependent of the price as linear problem in a consistent manner

Another advantage of the Boltzmann model is a fact that a product (or aflux) of the probability density function and the price change rate perunit time for the financial product is introduced for applying avariance reduction method in order to improve the computationefficiency.

Introducing the concept of flux makes it possible to simulate a pricefluctuation for a financial product at arbitrary time. To be moreprecise, the probability of the price for the financial product isevaluated from the estimated track-length of the flux for the financialproduct, which is just like the track-length estimator in the neutrondiffusion equation. This arrangement is capable of reducing the variancein the Boltzmann analysis using a Monte Carlo method.

The Boltzmann model also introduces the concept of point detector usedin the neutron transport calculation, in addition to the concept of theflux, into evaluation of price fluctuation for the financial product orthe derivatives. This arrangement can also reduce the variance of theBoltzmann model analysis using a Monte Carlo method.

For example, using all of or a part of the events of price fluctuationfor the financial product to evaluate the probability at aninfinitesimal time band and an infinitesimal price band, the variancecan be reduced. No events of price change are likely to occur or nofluxes can pass through in such an infinitesimal band during the randomsampling, or no fluxes can pass.

Another method for reducing the variance is to calculate an adjointprobability density and an adjoint flux of the price change for thefinancial product or the derivatives, and weight the sampling inproportion to the adjoint probability density and the adjoint flux. Thismethod is also based on the introduction of the concept of the flux.

The Boltzmann model is applicable to multiple financial products or thederivatives that requires consideration of the correlation among theseproducts.

When evaluating a portfolio that is financial derivative productconsisting of a combination for multiple financial products, thecorrelation among the financial products or their derivatives are takeninto account, the Boltzmann model can be adopted to the conventionalevaluation system.

In the field of financial engineering, the Ito's theorem is well-known.According to the Ito's theorem, if a financial product obeys the Ito'sprocess, then the derivative product also obeys the Ito's theorem. TheIto's theorem is applicable to the Boltzmann model as well. Accordingly,when evaluating a price or a risk for a financial derivative product inthe Boltzmann model, the price distribution of the derivative productcan be propagated by the original underlying assets, based on the Ito'stheorem. Thus, technique of the present invention can be applied toconventional price and risk evaluation systems. As a result, theBoltzmann model can be employed in the conventional systems.

Finally, the Boltzmann model of the present invention carries outsimulations for price fluctuation, and accumulates the probabilitydistribution of individual simulation to obtain the price distributionand the risk distribution. Accordingly, simulations of price fluctuationcan be executed in the parallel manner by the price fluctuationsimulator 15 and the probability density computation unit 16 to improvethe operation speed.

The foregoing is the theoretical background of introducing the Boltzmannmodel for risk analysis of the financial product. Subsequently, theprice distribution and the risk distribution for the financial productby the Boltzmann model will be explained in detail using actual examplesof analysis results.

FIG. 2 illustrates the operation flow of the Boltzmann model analysisunit 5. The Boltzmann model analysis unit 5 executes the steps A throughI, as shown in FIG. 2.

Step A is initialization of the Boltzmann equation. The initializationunit 12 executes this step.

Step B gives initial values of Boltzmann equation. Solving the Boltzmannequation is equivalent to obtaining the Green's function of theBoltzmann equation having a source term. The Green's function representsvirtual particle diffusion distribution in the phase space from a pointsource. By setting the initial values x₀, v₀, and μ₀ for the price x,the price change rate v, and the price change direction μ, the sourceterm of the Boltzmann equation is given. This step is carried out by theinitial value setting unit 13.

Step C determines a sampling method. In this example, it is assumed thatthe price changes once a day. Sampling in accordance with the frequencyof price change based on the total cross-section will be describedlater. The sampling unit 14 executes step C.

Step D simulates price fluctuation using the Monte Carlo method.The Integral $\begin{matrix}{\xi_{i,2} = {\int_{0}^{v_{i} + 1}{{{\mathbb{d}v} \cdot v}\quad{\exp\lbrack {- \frac{v}{T}} \rbrack}}}} & (16)\end{matrix}$integrates formula v·exp[−v/T], which form has similarity with anempirical form for neutron kinetic energy emitted in fission reactionwith the velocity term in the differential cross-section. In otherwords, price fluctuation is simulated by giving a velocity distributionto the Boltzmann equation, and by calculating v_(i+1) satisfying theprobability associated with random numbers having been generated. Theprice fluctuation simulator 15 carries out step D.

In the example shown in FIG. 2, the price change direction is simulatedin such a manner that the price-up direction μ_(i+1) equals 1(μ_(i+1)=1)and the price-down direction μ_(i+1) equals −1(μ_(i+1)=−1) simply fromthe value of the random number.

Step E carries out integration of the Green's function to obtain theprobability density Pm. The probability density computation unit 16carries out step E.

Step F determines whether or not one trial has been completed. Theone-trial completion detector 17 executes step F. If one-trial has notbeen completed yet, the process returns to step C.

Steps G and H determine whether or not all trials have been completed.If all trials have not been completed yet, the process returns to stepB. The all-trial completion detector 18 carries out steps G and H.

Step I edits the calculated probability density, which is carried out bythe probability density editor 19.

In the example shown in FIG. 2, sampling is conducted on the assumptionthat a price change occurs once a day. FIG. 3 illustrates anotherexample, in which the sampling interval is set in response to thefrequency of price change.

The process shown in FIG. 3 is almost the same as that shown in FIG. 2,except for steps C′ and E′. In step C′, the sampling interval set inaccordance with the frequency of price change. The microscopic totalcross-section in the Boltzmann equation means the inverse of the meanfree path (which is the average distance from one collision to the nextcollision) of the neutron. A collision frequency (which is the collisionprobability per unit time) is the product of the total cross-section andthe velocity v. By applying this to the change of a stock price,simulation of price fluctuation can be carried out without setting atime grid, as long as the stochastic process of the change in the stockprice and the total cross-section are known.

In the conventional technique, using Δt as a certain change in time, theprice after Δt is simulated by a normal random number having a standarddeviation of σ√Δt. For this reason, the time grid Δt must be set as anindispensable step for conducting an accurate simulation. On thecontrary, with the price and risk evaluation system of the presentinvention, setting of the time grid can be omitted in step C′.

In step C′ shown in FIG. 3, the index distribution used in thesimulation of price fluctuation is expressed as $\begin{matrix}{\tau_{i} = {\frac{{- \ln}\quad( {1 - \xi_{i,1}} )}{v_{i}{\Sigma_{t}( v_{i} )}}.}} & (17)\end{matrix}$

In step C′, the stochastic process and the total cross-section aresupplied from the all cross-section/stochastic process input unit 7.

In step D of FIG. 3, price fluctuation is simulated in accordance withthe sampling method mentioned above. The price fluctuation simulationitself is substantially the same as the simulation of step D shown inFIG. 2.

The difference is that the sampling interval changes in response to thefrequency of price change in the process shown in FIG. 3. The samplinginterval is adjusted by determining whether the next sampling positionresides within the observation area (Am, Bm, Cm, Dm) after every pricesimulation.

FIGS. 4 and 5 schematically show the simulation within a predeterminedobservation area corresponding to the processes of FIGS. 2 and 3,respectively. Especially, it is clearly shown in FIG. 5 that the pricefluctuation can be simulated at a specific time interval depending onthe frequency of price fluctuation, irrespective of the time grid.

The simulation method of price fluctuation based on the Boltzmann modelaccording to the present invention can also carry out simulation basedon the conventional diffusion model if equation 18 is used as a functiondescribing time T, direction M, and velocity V. $\begin{matrix}{{{T( {\xi_{{3i} + 1},x_{i - 1},v_{i - 1},\mu_{i - 1}} )} = {- \frac{\ln\quad( {1 - \xi_{{3i} + 1}} )}{\lambda}}}{{M( {\xi_{{3i} + 2},x_{i - 1},v_{i - 1},\mu_{i - 1}} )} = \begin{matrix}1 & {( {{{if}\quad\xi_{{3i} + 2}} \geq 0.5} ),} \\{- 1} & ( {{{if}\quad\xi_{{3i} + 2}} < 0.5} )\end{matrix}}{{V( {\xi_{{3i} + 3},x_{i - 1},v_{i - 1},\mu_{i - 1}} )} = \begin{matrix}{v_{0} + ( {r - \frac{\sigma^{2}}{2}} )} & {( {{{if}\quad\mu_{i - 1}} = 1} ),} \\{v_{0} - ( {r - \frac{\sigma^{2}}{2}} )} & ( {{{if}\quad\mu_{i - 1}} = {- 1}} )\end{matrix}}} & (18)\end{matrix}$

The results of two examples will be shown below.

EXAMPLE 1 r=0.05, σ²=0.11, and T=0.25 Year

The lower and upper limits Am and Bm of x (price) shown in the flowchartof FIG. 3 define a price band, which is determined by dividing the rangeof −3σ>x>3σ(δx=0.1σ) by 60. The lower limit Cm and the upper limit Dm oftime t define a time band, where Cm equals 0.25 [years] and Dm equalsCm+(1/365) [years]. An evaluation quantity ω_(i) is 1.

FIG. 6 shows the evaluation result of example 1 using the solid line 22in comparison with the theoretical distribution (i.e., the logarithmicnormal distribution) indicated by the dashed line 21. As is clear fromFIG. 6, the simulation result of the present invention indicated by thesolid line 22 is almost coincident with the theoretical distribution 21.

EXAMPLE 2 σ²=0.1, r=0.05 and r=0

The lower and upper limits Am and Bm of x shown in the flowchart of FIG.3 are set to Am=−∞and Bm=+∞. The time band defined by Cm and Dm are from0 to 365 [day] with δ_(t)=1 [day]. The evaluation quantity ω_(i) is x.In FIG. 7, the dashed line 23 indicates the theoretical distribution 23under a drift, and the long dashed line 24 indicates the theoreticaldistribution 24 without a drift. The simulation results 25 and 26obtained in example 2 substantially reproduce the theoreticaldistributions with and without a drift.

The velocity distribution and the direction distribution used in theabove-explained simulation are the probability distribution same as thestandard Gaussian distribution. Consequently, the probability densityobtained by the simulation becomes equivalent to that of the diffusionmodel.

In order to realize the Boltzmann model, the velocity distribution andthe direction distribution must be evaluated.

An example of how to evaluate the velocity distribution will be nowexplained. To apply the geometric Brownian motions, the naturallogarithm of stock price is defined as x. This corresponds to theposition x of a neutron.

From stock prices of about sixty Japanese electric machinery makers overthree years, the velocity distribution and the direction distributionare evaluated. To define x, the natural logarithm of the closing price(or the last price) of each day is input to x. An incident velocity v′is defined as the absolute value of the difference between naturallogarithm of the closing price of the current day and natural logarithmof the closing price of the previous day. A current velocity v isdefined as the absolute value of the difference between the naturallogarithm of the closing price of the current day and the naturallogarithm of the closing price of the next day.

The incidence direction μ′ is represented by the negative or positivesign of v′, and the current direction μ is represented by the negativeor positive sign of v. Since a change per day is observed, thedeterministic drift term (e.g., a non-risky interest rate) is omitted.If any drifts are found in the simulation using the actual datamentioned above, it is a purely stochastic drift.

In order to obtain the velocity distribution of the stock price, thespectrum shown in FIG. 8 is required. The spectrum is the integral of x,μ, t of the density p (x, v, μ; t), and is expressed by equation (19).S(v)=∫dtdμdx·p(x,v,μ;t)  (19)

In FIG. 8, the darkened circle (●) indicates the total spectrum 27expressed by equation (19). The spectra of the negative (or price-down)direction and the positive (or price-up) direction are indicated bystars (*) and white squares (□) 29, respectively. The price-downspectrum S⁻(v) and the price-up spectrum S₊(v) are expressed asS ⁻(v)=∫dtdx·p(x,v,−1;t)S ₊(v)=∫dtdx·p(x,v,1;t)  (20)

Although the two spectra are slightly different in size, the shapes arethe same. This result shows that the velocity distribution and thedirection distribution are independent of each other. Thesedistributions are described by the sigmoid function expressed byequation 21 and its approximation using the Maxwell distribution as atypical example. $\begin{matrix}{{f(v)} \propto \frac{v^{\kappa}\zeta\quad g\quad{\exp\quad\lbrack {\gamma\quad v} \rbrack}}{{g\quad{\exp\quad\lbrack {\gamma\quad v} \rbrack}} + \zeta - g}} & (21)\end{matrix}$here, v, k, ζ, g, and γ are arbitrary real numbers.

These spectra are approximated as indicated by the steep slop 30 and thegentle slop 31. In other words, the two spectra correspond to the twocomponents, namely, the steep slope component and the gentle slopecomponent. The curve 32 indicates the Gaussian distribution. TheGaussian distribution almost reproduces the steep slope component, butit evaluates the gentle slop component excessively small.

FIG. 9 exhibits the dependency of the spectra on the incident velocity.The darkened marks 33, the cross marks 34, and the white marks 35represent the velocity distributions with the incident velocities ofabout 1%, about 2%, and about 3%, respectively. These distributions arenormalized to 1.0 with integration.

FIGS. 10 and 11 illustrate the double differential cross-section Σ(V′,μ′→v, μ) with respect to direction μ. From FIGS. 10 and 11, it isapparent that the shapes of the spectra are the same. This factindicates that the double differential cross-section is given by theproduct of the velocity distribution ℑ(v′→v) and the directiondistribution

(μ′→μ). This can be expressed by equation 22.Σ(v′,μ′→v,μ)∝ℑ(v′→v)

(μ′→μ)  (22)

Here is an example of determining the distribution of the price changeby a sigmoid function, using the past records for price change rate as aparameter. As illustrated in FIGS. 9 through 11, the spectra shift tohigher values as the incident velocity v increases. Therefore, theconcept of temperature can be introduced, and the distributions shown inFIGS. 9 through 11 can be described by the Maxwell distribution.

Since the distributions shown in FIGS. 9 through 11 suggest anexponential distribution, the evaporation spectrum used to describe theneutron emitted from a nuclear reaction, which is expressed by equation(23), will be used. $\begin{matrix}{{f(v)} \propto {v\quad{\exp\quad\lbrack {- \frac{v}{T}} \rbrack}}} & (23)\end{matrix}$

This is a modification of the Maxwell distribution. The naturallogarithm of equation 23 becomes $\begin{matrix}{{{\ln( {f(v)} )} - {\ln(v)}} = {{- \frac{v}{T}} + {{Const}.}}} & (24)\end{matrix}$

FIG. 12 illustrates the relationship of equation (24). The inverse ofthe slope corresponds to temperature T. FIG. 12 gives an experimentalequation of differential cross-section expressed by equation (25), andFIG. 13 illustrates the relationship between the velocity v′ andtemperature T. $\begin{matrix}{{{\mathfrak{J}}( v^{\prime}arrow v )} \propto {v\quad{\exp\lbrack {- \frac{v}{{3.5533\quad v^{\prime 2}} + {0.1023\quad v^{\prime}} + 0.0044}} \rbrack}}} & (25)\end{matrix}$

Next, an example of estimating the probability of the change directionfor a financial product or the derivatives from the past records of thatfinancial product will be explained.

In this problem, the direction takes values of only 1 and −1. The value“1” denotes increase in price, and “−1” means decrease in price. In thefinancial engineering, the direction distribution is given by equation(26). $\begin{matrix}{{\wp( { \mu^{\prime}arrow\mu ;t} )} = \{ \begin{matrix}{\wp( { 1arrow 1 ;t} )} & ; & {{continuously}\quad{price}\text{-}{up}} \\{\wp( { {- 1}arrow 1 ;t} )} & ; & {{change}\quad{from}\quad{price}\text{-}{down}\quad{to}\quad{price}\text{-}{up}} \\{\wp( { 1arrow{- 1} ;t} )} & ; & {{change}\quad{from}\quad{price}\text{-}{up}\quad{to}\quad{price}\text{-}{down}} \\{\wp( { {- 1}arrow{- 1} ;t} )} & ; & {{continuously}\quad{price}\text{-}{down}}\end{matrix} } & (26)\end{matrix}$

FIG. 14 illustrates the averages of the continued price-up probability

(1→1; t) and the continued price-down probability

(−1→−1; t) of every five days. The darkened squares (▪) 36 represent theevents transient from price-up to price-up, the probability of which isexpressed by

(1→1; t), and white squares (□) 37 represents the events transient fromprice-down to price-down, the probability of which is expressed by

(−1→1; t). The bold horizontal solid line 38 and the dashed line 39 arethe time averages of these two probabilities. Other two probabilitiesare expressed by equation (27).

(1→−1;t)=1−

(1→1;t)

(−1→1;t)=1−

(−1→−1;t)  (27)

FIG. 14 exhibits the correlation between the probability of price-up andthe probability of price-down with respect to the probability of thechange direction for a financial product or its derivatives. FIG. 14clearly shows that probability of price-up (denoted by ▪36) and theprobability of price-down (denoted by □37) change in opposite directionsas time passes. This fact indicates a negative correlation.

By reflecting the correlation shown in FIG. 14 into the angledistribution of the double differential cross-section, a more accurateevaluation of the stochastic drift can be achieved.

Next, comparison will be made between the evaluation results with theprice and risk evaluation system of the present invention and theevaluation results with the conventional technique.

FIG. 15 shows the evaluation results with the Boltzmann model that usesthe price change rate distribution and the price change directiondistribution. The solid lines 40 represent the results from theBoltzmann model, which effectively reproduce the jumps (big changes) inprice appearing in the thick line 41 that indicates the actual record.

FIG. 16 is a detailed view of FIG. 15. The track depicted by thedarkened squares 42 is the real record. The real record exhibits severalbig changes (jumps) 43 of about 10% per day.

The white squares (□) 44 and the white triangles (Δ) 45 are obtainedfrom the simulation with the Boltzmann model. These symbols exhibitjumps 46 that are similar to the jumps 43 in the real record 42. Theability of simulating price jumps is the significant feature for theprice and risk evaluation system of the present invention.

In contrast, the conventional diffusion model (denoted by cross marks 47in FIG. 16) is incapable of reproducing the abrupt jumps, and itsimulates price fluctuation only in the continuous manner.

FIG. 17 illustrates the simulation result obtained by the Boltzmannmodel in comparison with logarithmic distribution of a stock price ofafter 200 days. In this simulation, the deterministic drift term (forexample, a non-risky interest rate) is not taken into account. In FIG.17, the solid curve 48 indicates the conventional diffusion model (withthe expectation value of 0 and the standard deviation of 0.29). Thedarkened squares 49 indicate the Boltzmann model, which is slightlydrifted due to purely the stochastic process. The dashed line 50indicates the corrected Gaussian distribution as a result of correctionof the drift term. The corrected Gaussian distribution 50 covers a rangeof about ±2.5σ of the Boltzmann model; however, the other portions wereunderestimated.

The simulation result shown in FIG. 17 exhibits the significant featureof the Boltzmann model well. That is, the Boltzmann model used in thepresent invention is capable of evaluating the stochastic drift and theprobability of big jumps in price, which can not be systematicallyreproduced by the conventional technique.

Next, another application will be explained with a case in which acorrelation between the velocity v (corresponding to the price changerate) and the angle μ (corresponding to the price change direction)exists. In the above-described examples, the velocity v and the angle μare supposed to be independent of each other with respect to the doubledifferential cross-section, and therefore, separation form of variableswas applied, as expressed in equation (22).

However, as the significant feature, the Boltzmann model is capable oftaking the correlation between the velocity v and the angle μ intoaccount by introducing a function that is not subjected to separation ofvariables. The above-described examples without considering thecorrelation only evaluate symmetrical distributions, which are symmetricwith respect to the mean value. On the contrary, in this example, anasymmetric distribution can be evaluated by taking the correlationbetween the probability of the price change rate and the probability ofthe price change direction, and therefore, by introducing a function ofnon-separation of variable.

Next, a heterogeneous problem will be explained. In equation (7), thecross-section Σ is constant with respect to the price x. This is thesame thing as the conventional financial engineering in which thevolatility is constant with respect to the price. When the conventionalfinancial engineering treats a heterogeneous problem containing aninconstant cross-section Σ, the price-dependency of the volatility hadto be corrected by a technique of volatility smile or other techniques.However, these techniques greatly rely on past experiences and know-how.In contrast, the Boltzmann model applies equation (6) expressing aheterogeneous problem, which allows the price-dependency of volatilityto be systematically considered.

If the heterogeneity varies due to the change in the price distribution,a non-linear Boltzmann equation will be introduced. Since theconventional financial engineering is not able to theoretically treatheterogeneous problems, these problems are often treated as stochasticvolatilities in the conventional technique. However, stochasticvolatility itself greatly relies upon know-how or experiences, and islack of objectiveness. On the contrary, the Boltzmann model used in thepresent invention can treat stochastic volatility accurately using anon-linear Boltzmann equation (28). $\begin{matrix}{{- \frac{\partial{\phi( {x,v,{\mu;t}} )}}{v{\partial t}}} = {{\mu\frac{\partial{\phi( {x,v,{\mu;t}} )}}{\partial x}} + {{\Sigma_{t}( {{\phi( {x,v,{\mu;t}} )},v} )}{\phi( {x,v,{\mu;t}} )}} - {\int{{\mathbb{d}v^{\prime}}{\mathbb{d}\mu^{\prime}}{\Sigma_{s}( {{\phi( {x,v,{\mu;t}} )},v^{\prime}, \mu^{\prime}arrow v ,\mu} )}{\phi( {x,v^{\prime},{\mu^{\prime};t}} )}}}}} & (28)\end{matrix}$

In equation (28), flux φ is contained in the cross-section, and thecross-section changes along with the change in flux. This technique isactually used in computation of burning of nuclear fuel in theneutron-relating field. By applying this technique to the financialengineering, a systematic evaluation method for stochastic volatility isrealized.

Next, the evaluation at an arbitrary point of time within an observationarea will be explained. The density expression in equation (10) can notevaluate the probability density of an arbitrary hour between the 199thday and the 200th day because the event of price change can not bedetected.

However, if the flux expressions of equations (6) and (7) are used, fluxφ(x, v, μ; t) is obtained at an arbitrary time t irrespective ofpresence or absent of price change. Since flux φ(x, v, μ; t) candescribe an arbitrary time t, and therefore, the probability density Pat an arbitrary hour between the 199th day and the 200th day can beobtained correctly using equation 29. $\begin{matrix}{P = {\int_{{\Delta\quad t} = 1}{{\mathbb{d}t}\frac{\phi( {x,v,{\mu;t}} )}{v}}}} & (29)\end{matrix}$

This is a track-length estimator.

Next, a method, for making evaluation at a point that does not alloweffective sampling, will be explained. Although the estimation of thetrack length allows evaluation at an arbitrary time, it is impossiblefor the conventional technique to effectively collect sampling data atan infinitesimal price band and an infinitesimal time band even ifintensive sampling is carried out. This means that evaluation can not bemade from the estimation of track length.

In the present invention, the concept of point detector for evaluating aneutron at an arbitrary point in the phase space is applied to theevents for financial products in order to allow evaluation at aninfinitesimal interval. With the concept of point detector, a neutronthat reaches point C is evaluated by computing the probability that theneutron starting from point A collides at point B and is scatteredtoward point C. The probability that the neutron passes point C isestimated from scattering information that the neutron does not passpoint C. In the simplest example, with the distance r between points Band C, the neutron decays by exp (−Σ_(t)·r), and the solid angle changesin accordance with the distance. In this case, the neutron that changesat point B and reaches point C is estimated accurately to theconsiderable extent by correction of 1/r². Because it is known that theprobability of the scattering angle at point B is the differentialcross-section Σ_(s)(v′, μ′→v, μ), the probability of the neutron thatchanges at point B during the sampling and does not reach point C can beestimated.

The concept of point detector is introduced into evaluation for afinancial product, while using all of or a part of the events of theprice change for the financial product or its derivatives. Thisarrangement allows the price distribution or the risk distribution forthe financial product in an infinitesimal observation area (or a targetarea).

In reality, no events of price change are likely to occur during therandom sampling, and no flux pass through in an infinitesimal price bandand at an infinitesimal time. In spite of the fact that an event can notoccur in an infinitesimal area in the target phase space, theprobability of events in such an infinitesimal area can be evaluated bythe present invention within realistic computation time by automaticallychecking the route of causing the events. Evaluating the probability inan infinitesimal area, through which flux can not pass, allows thevariance to be reduced efficiently.

Next example shows a case in which an adjoint probability density oradjoint flux of the Boltzmann model is introduced. The adjoint equationof equation (7) is expressed by equation (30). $\begin{matrix}{{- \frac{\partial{\phi^{*}( {x,v,{\mu;t}} )}}{v{\partial t}}} = {{\mu\frac{\partial{\phi^{*}( {x,v,{\mu;t}} )}}{\partial x}} + {{\Sigma_{t}(v)}{\phi^{*}( {x,v,{\mu;t}} )}} - {\int{{\mathbb{d}v^{\prime}}{\mathbb{d}\mu^{\prime}}{\Sigma_{s}( {v, \muarrow v^{\prime} ,\mu^{\prime}} )}{\phi^{*}( {x,v^{\prime},{\mu^{\prime};t}} )}}} - {S^{*}( {x,v,{\mu;t}} )}}} & (30)\end{matrix}$The adjoint flux φ*(x, v, μ; t) is a solution of equation (30), and itrepresents the sensitivity of the expectation value of the a jointradiation source S*(x, v, μ; t). The expectation value is expressed byequation (31)

S*(x,v,μ;t)φ(x,v,μ;t)

=∫dvdμdxdtS*(x,v,μ;t)φ(x,v,μ;t)  (31)

The adjoint source S*(x, v, μ; t) corresponds to the price evaluationequation for a financial product or its derivatives.

By providing weights proportional to the adjoint flux φ*(x, v, μ; t) inthe phase space, the variance inevitably accompanying the Monte Carlomethod can be reduced when evaluating the expectation value of thefinancial product or its derivatives.

Next, an application will be made to the case in which there is acorrelation among financial products. For example, there is acorrelation expressed by equation (32) among multiple (e.g., two)financial products and their derivatives.dx ₁=μ₁ dt+σ ₁ √{square root over (dt)}ξ ₁dx ₂=μ₂ dt+σ ₂ √{square root over (dt)}ξ ₁  (32)If there is the correlation of equation (32) among the financialproducts, the conventional technique simulates a price by generating acorrelative random number in accordance with a known correlationcoefficient when producing normal random numbers ξ₁ and ξ₂ with respectto two Ito's processes. This conventional method is capable ofevaluating not only the price for a single financial product, but alsothe price of a portfolio consisting of a combination of multiplefinancial products.

In contrast, the present invention realizes application to the portfolioby simultaneously setting multiple equations (33) for multiple financialproducts based on the Boltzmann model. $\begin{matrix}{{- \frac{\partial{\phi_{1}( {x_{1},v_{1},{\mu_{1};t}} )}}{v_{1}{\partial t}}} = {{{\mu_{1}\frac{\partial{\phi_{1}( {x_{1},v_{1},{\mu_{1};t}} )}}{\partial x_{1}}} + {{\Sigma_{t}( v_{1} )}{\phi_{1}( {x_{1},v_{1},{\mu_{1};t}} )}} - {\int{{\mathbb{d}v_{1}^{\prime}}{\mathbb{d}\mu_{1}^{\prime}}{\Sigma_{s}( {v_{1}^{\prime}, \mu_{1}^{\prime}arrow v_{1} ,\mu_{1}} )}( {{\phi_{1}( {x_{1},v_{1}^{\prime},{\mu_{1}^{\prime};t}} )} + {\phi_{2}( {x_{1},v_{1}^{\prime},{\mu_{1}^{\prime};t}} )}} )}} - \frac{\partial{\phi_{2}( {x_{2},v_{2},{\mu_{2};t}} )}}{v_{2}{\partial t}}} = {{\mu_{2}\frac{\partial{\phi_{2}( {x_{2},v_{2},{\mu_{2};t}} )}}{\partial x_{2}}} + {{\Sigma_{t}( v_{2} )}{\phi_{2}( {x_{2},v_{2},{\mu_{2};t}} )}} - {\int{{\mathbb{d}v_{2}^{\prime}}{\mathbb{d}\mu_{2}^{\prime}}{\Sigma_{s}( {v_{2}^{\prime}, \mu_{2}^{\prime}arrow v_{2} ,\mu_{2}} )}( {{\phi_{1}( {x_{2},v_{2}^{\prime},{\mu_{2}^{\prime};t}} )} + {\phi_{2}( {x_{2},v_{2}^{\prime},{\mu_{2}^{\prime};t}} )}} )}}}}} & (33)\end{matrix}$

By considering the correlation between the double differentialcross-sections Σ_(s) (v₁′, μ₁′→v₁, μ₁) and Σ_(s) (v₂′, μ₂′→v₂, μ₂) forthe third terms of the right-hand-side of the respective equations, aneffect equal to or higher than the conventional technique using equation(32) can be achieved.

Next, the application of Ito's theorem into the Boltzmann model of thepresent invention will be explained. Ito's theorem defines that if theprice S of a financial product, such as a stock, obeys the Ito's processexpressed by equation (34), then the price F (S, t) of the derivativeproduct moves in accordance with the stock price, and also obeys theIto's process.dS=a(S,t)dt+b(S,t)√{square root over (dt)}ξ  (34)

In this case, the price F of the derivative product is expressed byequation (35). $\begin{matrix}{{dF} = {( {\frac{\partial F}{\partial t} + {a\frac{\partial F}{\partial S}} + {\frac{1}{2}b^{2}\frac{\partial^{2}F}{\partial S^{2}}}} ) + {b\frac{\partial F}{\partial S}\sqrt{dt}\xi}}} & (35)\end{matrix}$

The conventional technique that does not apply the Ito's theorem isbased on the assumption that the random number ξ, has a normaldistribution. On the contrary, the Ito's theorem stands in the Boltzmannmodel even if the distribution is not Gaussian, as long as the randomprocess of the second term is proportional to the square root of theinfinitesimal time dt.

For this reason, the Boltzmann model applying the Ito's theorem canevaluate a price distribution irrespective of whether or not thedistribution is Gaussian. If the variance of the distribution evaluatedby the Boltzmann model is proportional to the square root of time, theIto's theorem can be applied to a conventional price/risk evaluationsystem. In this case, the conventional system applying the Ito's theoremwill achieve the similar effect as the present invention by replacingthe normal distribution of the random number x with the distributionobtained by the Boltzmann model.

Although a price distribution obtained by the Boltzmann model isdifferent from a price distribution using Gaussian in the strict sense,the distribution obtained by normalizing the price to the standarddeviation become constant independent of time, as shown in FIG. 18. InFIG. 18, the darkened square (▪) 51, the darkened triangle (▴) 52, andthe darkened diamond (♦) 53 show the price distribution of after 20days, after 60 days, and after 100 days, respectively. The white square(□) 54, the white triangle (Δ) 55, and the white diamond (

) 56 show the price distribution of after 120 days, after 160 days, andafter 200 days, respectively. These data coincide with the normaldistribution 57 within the range of about ±2.5σ. The standard deviationof this normal distribution is proportional to the square root of time.This means that the standard deviation of the probability distributionobtained by the Boltzmann model is in proportion to the square root oftime.

The example shown in FIG. 18 is the simulation result of the stock-pricedistribution for about sixty Japanese electric machinery makers. If aconventional system is designed based on the Ito's theorem, it canevaluate a price distribution or a risk distribution for derivativesderiving from the stock prices of these makers. In this case, theconventional system using the Ito's theorem can replace the normaldistribution with the Boltzmann model distribution in order to improvethe prediction ability for a price distribution of the derivatives.

Lastly, an application of the present invention to a parallel processingsystem will be explained. The present invention uses the Monte Carlomethod for the numerical calculations. It is widely known that the MonteCarlo method is an advantageous technique because the processing speedcan be drastically improved by parallel processing. Especially, it iswell known that parallel processing is quite effective in application ofthe neutron transport Monte Carlo method. Since the present inventionmakes use of the neutron transport Monte Carlo technique in theBoltzmann model, the calculation speed can be effectively improved usingparallel processing computers.

FIG. 19 illustrates an example of a parallel processing system. In FIG.19, each operation flow A through I is the same as that shown in FIGS. 2and 3. Especially, simulation process consisting of steps B through F isdivided into multiple parallel flow in order to allocate the trials to aplurality of CPUs. The operation speed increases depending on the numberof the CPUs used in the parallel processing.

The foregoing is the preferred embodiment of the first feature of thepresent invention, which is realized as a price and risk evaluationsystem for a financial product. The present invention is also realizedas a computer-readable recording medium storing the price and riskevaluation program, which controls a computer system to carry out theprocess described above. In this case, the program is installed in acomputer system, and the price and risk evaluation system is realizedwhen the program is started on the computer system.

The second feature of the present invention, which is embodied as adealing system, will be described with reference to FIG. 20 and thesubsequent drawings. First, option-pricing methodology will be explainedin the frame work of the theoretical and realistic aspects. A price ofEuropean option for stocks and stock indices at delivery dates can beevaluated using integral equations (36) and (37) based on therisk-neutral probability measure (probability density) P (S, τ) forunderlying assets in the free market.Call(K,τ)e ^(−rτ)∫_(K) ^(∞) dSP(S,τ) (S−K)  (36)Put(K,τ)e ^(−rτ)∫₀ ^(K) dSP(S,τ) (K−S)  (37)here S is the price of the underlying assets, τ is the period to thematurity, r is the non-risky interest rate (that is, the money ratefixed up to the maturity), and K is the exercise price.

Equation (36) expresses the theoretical call option price to buy theunderlying assets (i.e., call option) at the maturity with the exerciseprice K. Equation (37) expresses the theoretical put option price tosell the underlying assets (i.e., put option) at the maturity with theexercise price K. A purchaser of these options can exercise the right atthe exercise price K irrespective of the actual price of the underlyingasset at the maturity. For example, the purchaser of a call option canbuy that option at price K, even if the underlying price (i.e., theprice of the underlying assets) is higher than the exercise price at thematurity. The purchaser of a put option has an obligation of selling theoption at the maturity at price K. However, this purchaser canrepeatedly trade the underlying assets in response to price changes, andcan sell the option at price K at least at cost of equation (36).

Black-Sholes equation (BS equation) is often used to evaluate an optionprice. If the logarithm normal distribution expressed by equation (38)is input to the risk neutrality probability measurement in equations(36) and (37), then Black-Sholes equations (39) and (40) are obtained.$\begin{matrix}{{P( {S,\tau} )} = {\frac{1}{S\quad\sigma\sqrt{2{\pi\tau}}}{\exp\lbrack {- \frac{( {{\ln\quad(S)} - \frac{\sigma^{2}\tau}{2}} )^{2}}{2\sigma^{2}\tau}} \rbrack}}} & (38) \\{{{Call}\quad( {K,\tau} )} = {{{SN}( d_{1} )} - {K\quad{\mathbb{e}}^{{- r}\quad\tau}\quad{N( d_{2} )}}}} & (39) \\{{{Put}\quad( {K,\tau} )} = {{{- {SN}}\quad( {- d_{1}} )} + {K\quad{\mathbb{e}}^{{- r}\quad\tau}{N( {- d_{2}} )}}}} & (40)\end{matrix}$here d₁ and d₂ are expressed by $\begin{matrix}{{d_{1} = {{\ln\quad( \frac{S}{K} )} + {( {r + \frac{\sigma^{2}}{2}} )\frac{\tau}{\sigma\sqrt{\tau}}}}}{d_{2} = {d_{1} - {\sigma\sqrt{\tau}}}}} & (41)\end{matrix}$

The parameter σ in equations (38), (39) and (40) is the price changerate (or volatility), and is the diffusion constant of the geometricBrownian motion model, in which the underlying price diffuses withrespect to the logarithm of the price, for the underlying assets.

The Black-Sholes equation is derived on the assumption that thevolatility σ is constant with respect to τ and S. Accordingly, theBlack-Sholes equation assumes a statistic market that exhibits aconstant irrespective of time and price.

However, the real market changes as time and price change. FIG. 20illustrates the price change rate C1 for the underlying assets predictedby the geometry Brownian model, in comparison with the change rate ofthe closing price (i.e., the daily earning rate) C2 for a typical stockprice. Although the volatility of the two data are almost the same, theappearances of the price change quite differ from each other. Thegeometric Brownian motion model C1 does not exhibit a big price change,whereas the actual market price significantly varies as indicated by thecurve C2. This comparison result leads to the conclusion that it isdifficult to evaluate the option price based on the Black-Sholesequation, if the underlying assets is the individual stock price. Inreality, the transaction of individual stock option is small in number.

The stock index, that is, the corrected average of the stock prices ofmany issues (for example, the Nikkei 225 Stock Average) moves moremoderately than the individual stock price. Accordingly, it becomeseasier to evaluate the option price for the stock based on theBlack-Sholes equation, and many transactions are carried out at thepresent. However, even if taking Nikkei 225 stock average, theappearance of the daily earning rate C3 is still different from thegeometric Brownian motion model (curve C1 in FIG. 20), as shown in FIG.21. If curve C3 is compared with curve C2 of actual record shown in FIG.20, these two curves are essentially the same, except for the size inchange. To this end, dealers of stock index options generally usemodified Black-Sholes equations to evaluate option prices.

FIG. 22 illustrates an example of the correction. The implied volatility(IV) is defined as the volatility implying an option price actuallytraded in the market using the Black-Sholes model. The darkened squares(▪) M51 shown in FIG. 22 represent a typical implied volatility of theclosing price for a stock index put option of the Nikkei 225 StockAverage.

The horizontal line C4 extending at 30% of the vertical axis is thehistorical volatility calculated from the motion of the option of theNikkei 225 Stock Average. If the market completely obeys the geometricBrownian movement model that is the basis of the Black-Sholes equation,the darkened squares (▪) M51 should be located on the 30% line C4.However, in reality, as the exercise price separates from the underlyingprice, the implied volatility tends to increase. The tendency that theimplied volatility increases from the point at which the exercise priceand the underlying price are equal, that is, with the ratio of (exerciseprice)/(underlying price) being 1.00, is called a smile curve, which isindicated as C5 in FIG. 22. It is known that the smile curve has a termstructure in which the curvature becomes gentle as the term (or period)increases up to the maturity.

In general, option-dealers try to grasp the volatility matrix, whichbring the smile curve and the term structure of the implied volatilitytogether based on the transaction price in the market. They determinethe option price by correcting the option price obtained by theBlack-Sholes equation using the volatility matrix.

Although the volatility matrix is one of the most successful tools forevaluating an option price, it still has some drawbacks. The majordrawbacks of the volatility matrix are the following two:

1. If there is no or a few transactions, it is impossible to obtainimplied volatility.

2. The volatility matrix can not specify a typical transaction in themarket, in which the transaction price varies widely.

The drawback 1 is the essential problem concerning the impliedvolatility, and can not be solved by the implied volatility. On theother hand, the drawback 2 could be solved by a filtering technique forextracting significant information among from the widely variedinformation in order to specify the realistic transaction.

Prior to applying the filtering technique, the mechanisms, why the smileand the term structure of the implied volatility appear, must beclarified in order to grasp the average behavior. Although themechanisms of the smile curve and the term structure have not beencompletely clarified yet, various researches suggest that the majorreasons are relating to the Leptokurcity and the Fat-tail occurring inthe probability distributions for the actual price changes in free trademarkets. Leptokurcity is the phenomenon that the probability of theactual price changes observed is sharper than the normal distributionassumed in the Black-Sholes equation in the region of small pricechange. Fat-tail is the phenomenon that the probability of the actualprice changes widens toward the end in the region of big price changes.

FIG. 23 illustrates an example of these phenomena. The distribution ofthe white squares (□) M52, which represent the actual daily earningrates, becomes sharper than the normal distribution C6 near the center,and broadens towards the ends. FIG. 24 illustrates the probability ofprice change estimated from the Boltzmann model, in comparison with thelogarithmic normal distribution of the Black-Sholes equation. Under thisprice distribution, the width (that is, the volatility) of theprobability density distribution is smaller than that of the logarithmicnormal distribution C7 around the relative price of 1.0, as is indicatedby darkened square (▪) M53. The volatility of the probability densitydistribution becomes larger than that of the logarithmic normaldistribution C7 in the ranges of the relative price of above 2.0 andbelow 0.5. It is supposed that the price distribution (indicated bywhite squares (□) M54 approaches the normal distribution as time elapsesaccording to the central limit theorem. Accordingly, the peak portionand the skirt of the price distribution become very similar to thenormal distribution C8 as time passes. This is supposed to be the factorof emergence of the smile curve and the term structure. The similardiscussion is made in “John C. Hull, “OPTIONS, FUTURES & OTHERDERIVATIVES, Fourth Edition”, Prentice-Hall International Inc., 2000,chapter 17.

The Fat-Tail of the price-change distribution corresponds to the bigprice changes that occur in the real price fluctuations C2 and C3 shownin FIGS. 20 and 21. There are two models taking such big changes intoaccount, namely, a Jump model and a probability volatility model. Thejump model reproduces the Fat-tail independently in the stochasticprocess that is totally different from the normal distribution. In theprobability volatility model, the standard deviation of the normaldistribution (that is, the volatility) fluctuates with time. The jumpmodel is based on the assumption of discontinuous price changes, whilethe probability volatility model is essentially a non-linear problem.For this reason, either model is incapable of achieving the risk-neutralprobability measure uniquely. Consequently, equations (36) and (37) ofevaluating option prices can not be applied to these two models, whichis the major drawback.

In contrast, although the Boltzmann model proposed by the presentinvention covers the category of probability volatility model in a broadsense, a linear Boltzmann equation can reproduce the characteristics ofthe Leptokurcity and Fat-Tail. If the angle distribution of the linearBoltzmann equation is isotropic (that is, if (μ′→μ; t)=½ in equation(26)), the solution becomes risk-neutral and unique. Therefore, applyingthe Boltzmann model to evaluation of option prices allows the essentialtrend of the volatility matrix to be properly estimated.

One of the significant features of the Boltzmann model is that thepresent model can treat market-dependency of price fluctuation. Themarket-dependency means that a set of big price changes occurcoincidentally with certain time intervals. The price evaluation systemthat has been described above as the first feature of the presentinvention preferably recommends applying an evaporation spectrumequation (42), which is a modification of the Maxwell's distribution, asthe price distribution f(v) in order to taking Leptokurcity intoaccount. $\begin{matrix}{{f(v)} \propto {v\quad{\exp\lbrack {- \frac{v}{T( v^{\prime} )}} \rbrack}}} & (42)\end{matrix}$

The Boltzmann model treats the correlation between the price change ratein the underlying assets and the previous price change rate. TheBoltzmann model claims the existence of a definite market-dependencybetween the daily earning rate v′ of the previous day and the dailyearning rate v of the current day via temperature T as exemplified inEq.(42) in case focusing on the closing prices. FIG. 25 illustrates atypical example of the market-dependency. In FIG. 25, the darkenedsquares (▪) M55 represent the temperature obtained from the real recordsof the closing price. The curve C9 is a fitting line of the darkenedsquares with a quadratic function. The fitting line exhibits the factthat the temperature T has a quadratic tendency with respect to thedaily earning rate v′ of the previous day expressed by equation (43).T(v′)=T ₀(1+c ₀ v′+g ₀ v′ ²)  (43)

The quadratic dependency recalls a direct analogy to the instability ofthe stock market in a system with a positive feedback such that thespecific heat increases as the temperature rises.

The curve C5 extending along the real records (i.e., the darkenedsquares (▪) M51) in FIG. 22 exhibits a volatility smile. This volatilitysmile is obtained by evaluating the option prices of equations (36) and(37) based on the Boltzmann model and plotting the volatility of theBlack-Sholes equation that become equal to the evaluation result. FIG.26 illustrates the daily return rate C10 obtained by the Boltzmann modelin the simulation process for price evaluation. The curve C 10 very wellreproduces the daily earning rate (i.e., the darkened squares (▪) M56)of the Nikkei 225 Stock Average in the same term as the optiontransaction.

The daily earning rate shown in FIG. 26 shows a typical Fat-Tail.

Based on this daily earning distribution obtained from the Boltzmannmodel, a random number ξ is generated to simulate the tracks of theunderlying assets S using equation (44). $\begin{matrix}{\frac{dS}{S} = {{rdt} + \xi}} & (44)\end{matrix}$

The jump model described above does not take the market-dependency intoaccount, and it treats a big price variation as abrupt and discontinuousjumps. This jump model may appear to give the similar results as theBoltzmann model; however, the result is quite different from thoseobtained from the Boltzmann model.

FIG. 27 illustrates the implied volatility of the Boltzmann model incomparison with that of the jump model. The solid line C11 and thebroken line C12 show the result of Boltzmann model with the terms towardthe maturity being 40 days and 80 days, respectively. Another brokenline C13 and the dashed line C14 show the result of the jump model withthe terms toward the maturity being 40 days and 80 days, respectively.

The comparison result shows that the implied volatility of the jumpmodels C13 and C14 become larger than that of the Boltzmann models C11and C12, and that the curvature of the smile curves of the jump model issmaller than that of the Boltzmann model. This result, that is, theexcessive implied volatility of the jump model, also applies to thecomparison between the jump model and the actual past records. Becausein the jump model the size of a price change is not correlated at all,the central limit theorem affects earlier. This is due to the earlierdiffusion of price in a discontinuous model, such as the jump model. Inorder to obtain the same result as the Boltzmann model, it is necessaryfor the jump model to use a distribution having a larger probabilitydensity in the region of lower earning rate than the curve C10, therebyreducing the diffusion of price. However, the resultant distribution ofthe daily earning rate becomes quite different from the underlyingassets. In this manner, the jump model differs from the Boltzmannfundamentally.

The Boltzmann model is not particularly complicated, as compared withthe jump model. This is true from the comparison between the Boltzmannmodel and the simplest jump model, for instance, the Merton's complexjump model. The Merton's jump model uses a random number ξ of the normaldistribution, a random number η of the Poisson distribution, a standarddeviation σ of the normal distribution, the average size k of jumps, andthe probability λ of occurrence of jumps per unit time. Thus, theMerton's jump model express stochastic differential equation (45) of theunderlying assets S using two probability density functions, that is,the Gaussian and the Poisson, and three parameters. $\begin{matrix}{\frac{dS}{S} = {{{- \lambda}\quad{kdt}} + {{\sigma\xi}\sqrt{dt}} + \eta}} & (45)\end{matrix}$

In contrast, the Boltzmann model uses a single probability densityfunction, that is, the Maxwell's distribution, and three parameters. TheBoltzmann model is simpler using a less number of probability densityfunctions.

The foregoing is the brief description of the related theory and thereality of option pricing. A dealing system for evaluating option pricesbased on the Boltzmann model will be explained hereafter.

FIGS. 28 and 29 illustrate the structure of a dealing system 100 of thepreferred embodiment of the present invention. The dealing system 100comprises an implied volatility calculation unit 102, a Boltzmann modelcomputation engine (BMM) 103, an implied volatility (IV) filter 104, anda dealing terminal 105. The implied volatility calculation unit 102communicates with an external market database 101, and acquires marketdata to calculate implied volatility. The Boltzmann model computationengine (BMM) 103 has a structure shown in FIG. 29, and conducts optionprice evaluation based on the Boltzmann model. The implied volatility(IV) filter 104 converts the option price provided from the BMM 103 intothe implied volatility (IV). The dealing terminal 105 functions as agraphical user interface (GUI), and displays necessary information. Thedealing terminal 105 also outputs data as, for example, hard copies, andinputs data to the system 100.

FIG. 29 illustrates the structure of the Boltzmann model computationengine (BMM) 103. The BMM 103 comprises an initial value setup unit 3,an evaluation condition setup unit 4, a Boltzmann model analysis unit 5,a graphical user interface (GUI) 105, a total cross-section/stochasticprocess setup unit 7, a velocity/direction distribution setup unit 8,and a random number generator 9. The initial value setup unit 103 allowsthe initial values of at least a price, a price change rate, and a pricechange direction to supply to the BMM 103. The GUI 105 may be used incommon with the GUI shown in FIG. 28. The BMM 103 is connected to anexternal market database 101 in order to take necessary market data in.

The Boltzmann model analysis unit 5 includes an initializing unit 12, aninitial value setting unit 13, a sampling unit 14, a price fluctuationsimulator 15, a probability density calculation unit 16. It also has aone-trial completion detector 17, an all-trial completion detector 18, aprobability density editor 19, a price distribution calculating unit 20,and price converter 21. The simulator 15 conducts calculations of pricefluctuation based on the Boltzmann model.

The invention will not be limited to the exemplified systemconfiguration, in which the system 100 is installed in a single computerin the physical meaning. For example, the system 100 can be a dividedsystem, such as a client-server system for conducting operationsseparately. Preferably, the elements in the system correspond toprograms for causing the associated elements to execute the operations,or perform the functions, indicated in the blocks in the drawings.Accordingly, the dealing system 100 can be realized by installing adealing program for causing a single computer with a communicationfunction to perform these functions.

The initial value setup unit 3 inputs T₀, C₀, g₀ of equation (43) to theBoltzmann model analysis unit 5. These parameters are directed to theunderlying assets of a stock price or a stock index to be evaluated, andare obtained from the past records. Preferably, the initial value setupunit 3 retrieves information on the stock price or the stock from themarket database 101, and acquires the initial values of the price, theprice change rate, and the price change direction from the retrievedinformation. The evaluation condition setup unit 4 supplies evaluationconditions to the Boltzmann model analysis unit 5. The evaluationconditions include the number of trials, a time band, a price rangeconcerned, which are required by the Boltzmann model analysis unit 5 toconduct essential calculations.

The Boltzmann model analysis unit 5 is the center of the dealing system,and the most essential element. The structure of the Boltzmann modelanalysis unit 5 are almost the same as that of the price and riskevaluation system shown in FIG. 1, but new elements, a pricedistribution calculation unit 20 and a price conversion unit 21, areadded.

The price distribution calculation unit 20 calculates the pricedistribution based on the price change probability density of theunderlying assets edited by the probability density editor 19.

The price conversion unit 21 computes and outputs an option price, basedon the price distribution calculated by the price distributioncalculation unit 20.

The dealing terminal 105, which functions as a GUI, outputs theintermediate operation results and the final results of the process. Thedealing terminal 105 also outputs the price distribution of the optionto be evaluated. This terminal 105 has an input function for allowingthe user to input data by a pointing device, such as a keyboard or amouse. The dealing terminal 105 has an output functions, such asdisplaying information on the monitor screen, printing out as hardcopies, transferring data to other systems via a network, and writing into the memories, etc.

The market database 101 stores information as to option products. Inthis context, the term “database” includes the data systematicallystored in the database, means for retrieving the data, and the hardwarestoring and managing these.

The market database 101 may be dedicated to this system, oralternatively, it may be an existing external database, if there areany.

An evaluation method of stock index option price carried out by theabove-explained system will now be described.

FIG. 30 illustrates six steps A1 through A6. In the evaluation of stockindex option price, many transaction data are generally utilized. Thepast records including such transaction data are accumulated in themarket database 101. In step A1, the total cross-section/stochasticprocess setup unit 7 calculates an implied volatility using the pastrecords stored in the database 101.

The steps A2 and A3 illustrated in the left half of FIG. 30 show theconventional procedure for the sake of comparison. In A2, a generaltrend of a volatility matrix is determined from the implied volatilityobtained in A1, based on the experiences and intuition, or otherwise, onthe simple path average or a recurrent model. Then, in A3, a volatilitymatrix is determined. The conventional technique required arbitraryjudgments in step of A2.

In contrast, the dealing system 100 of the preferred embodiment receivesthe implied volatility at the Boltzmann model analysis unit 5 in stepA4, and determines the temperature parameters (i.e., the threecoefficients T₀, c_(o), g_(o) in equation (43)) of the Boltzmann model.These parameters must be determined so that the temperature of theBoltzmann model coincides with the implied volatility.

In step A4, it is determined wither the Boltzmann model temperatureagrees with the implied volatility. If they agree with each other, thenthe process proceeds to step A5, in which it is determined whether theoutcome of the Boltzmann model agrees with the daily earning rate of theunderlying assets. If they do not agree with each other, the processreturns to step A4, and the parameters are reselected. If they agreewith each other, then the Boltzmann model is compared with themarket-dependency of the underlying assets in step A6.

If, in step of A6, the Boltzmann model agrees with the market-dependencyof the underlying assets, the process proceeds to step A3, in which avolatility matrix is determined based on the outcome of the Boltzmannmodel. In reality, it is rare that an apparent market-dependency isobserved. Accordingly, the flow from A5 to A6 becomes the final decisionin many cases. If an apparent market-dependency is observed, and if theBoltzmann model temperature does not agree with this market-dependency,then the process returns to step A4 in order to reselect the parameters.

The Boltzmann model can not always explain the real market very well.Even if a volatility matrix can be described very well in step A4, itmay be contradict with the daily earning rate. In such a case, theagreement between the Boltzmann model and the daily earning rate may begiven up, and the process proceeds to step A3. If the Boltzmann modeldoes not agree with the implied volatility in step A4, it means that theactual market exceeds the limit of the Boltzmann model. In this case,the process returns to the conventional step A2, and the evaluation isleft to the judgement of dealers.

Next, how the dealing system 100 evaluates the option price of anindividual stock will be explained.

It is observed that the daily earning rates for the option estimatedfrom the past records using the Boltzmann model agree with thereproduced option price based on the underlying assets through theBoltzmann model well. This fact means that the Boltzmann model hascapability for estimating the option prices based on the past records ofthe underlying assets, without past option-transaction records.Accordingly, the method of the present invention is the most efficientmethod for evaluating the option price of an individual stock havinglittle transaction record at present time.

Basically, the Boltzmann model uses the same technique for evaluatingthe option price of an individual asset as that for evaluating theoption price of a stock index. In other words, the option prices ofindividual asset can also be evaluated by determining the threecoefficients T₀, c₀, g₀ of equation (43) for each asset.

FIG. 31 illustrates the application of the tendencies reproduced byequation (43) to individual asset. FIG. 31 illustrates the temperaturesT of various assets listed in the Tokyo Stock Exchange. Along thehorizontal axis, various industries (constructions, foods, chemicals,steels, electric appliances, finances, and services) are arranged in theincreasing order of the historical volatility of stock price. Solidlines C21, C22, . . . , C27 in FIG. 31 represent the temperature Tconverted from the historical volatility. In the drawing, the darkenedcircles (symbol ●) denote the temperature of the earning-ratedistribution of the current day with the daily earning rate of within 5%of the previous day. The cross marks (X) denote the temperatures withthe daily earning rate of between 5% to 10% of the previous day. Thewhite squares (□) denotes the temperature with the daily earning rate ofbetween 10% to 15% of the previous day.

FIG. 31 reveals that the temperatures of three ranges are almostproportional to the corresponding historical volatility such as C21,C22, . . . , C27. It is also found that the temperature T increases asthe daily earning rate of the previous day becomes larger. Concerningthe distributions of the three symbols, the separation between the groupof the white squares and the group of the darkened circles (⊙) is largerthan the separation between the group of the white squares (□) and thecross marks (X). This means that as the daily earning rate of theprevious day increases, not only the temperature, but also thetemperature rising rate increases. These observations suggest thattemperature T has the quadratic form with the previous day's returns asexpressed in equation (43).

FIGS. 32 and 33 illustrate examples of option price evaluation forindividual stocks. FIG. 32 shows examples of evaluation for calloptions. The horizontal axis denotes the ratio of exercise price to theunderlying price (i.e., the price of the underlying assets). Thevertical axis denotes the ratio of the call option price to theunderlying price. FIG. 33 shows examples of evaluation for put options.The horizontal axis denotes the ratio of exercise price to theunderlying price. The vertical axis denotes the ratio of the price ofput option to the underlying price. In FIGS. 32 and 33, the solid linesC31, C41 represent the evaluated results of the Boltzmann model with theterm to the maturity of 20 days. The one-dot broken lines C32 and C42represent the evaluated results of the Boltzmann model with the term tothe maturity of 40 days. The dashed lines C33 and C43, and the dottedlines C34 and C44 represent the results based on the Black-Sholesequation of the corresponding terms.

Here, temperature T is given in the following equation (46) from FIG. 31under the assumption that an underlying asset concerned has historicalvolatility with the magnitude of about 70%.T(v′)=0.007(1+15v′+300v′ ²)  (46)

The magnitude about 70% of historical volatility is almost doubled withthe volatility of the stock index like option of Nikkei 225 stockaverage. Although this value is slightly larger than the historicalvolatility for many stocks, the magnitude is still in the range ofrealistic values.

FIGS. 34 and 35 illustrate the implied volatility for a call option anda put option, respectively. These two graphs exhibit smile curves andthe term structures, as the smile curves and the term structure for theexample of the implied volatility for the stock index shown in FIG. 27.

FIG. 36 shows the operation flow in evaluating the option price for theindividual stock executed by the dealing system 100. The left half ofthe drawing is the conventional flow, as in FIG. 30.

In general, the real records of the option transaction for variousstocks are insufficient to determine the historical volatility as thereliable information for option dealing. The option transactions withadequate records for various assets are handled with the analogous waysto the option dealing of a stock index, like the Nikkei 225 stockaverage. In the following explanation, the flow of FIG. 36 is appliedmainly to the option dealing with little records for inactivetransactions.

With the conventional technique, the smile and the term structure of thevolatility are inferred from the experiences in evaluation of optionprices for stock indexes, based on the historical volatility of theassociated asset, in step B1. Then, in step B2, a volatility matrix isdetermined based on the smile and the term structure.

In contrast, the dealing system 100 of the preferred embodimentdetermines the temperature parameters of the Boltzmann model for thisasset in step B3. Then, it is confirmed whether the daily earning rateof the Boltzmann model agrees with that of the underlying assets. If thedaily earning rates do not agree with each other, the process returns tostep B3 to reselect the parameters. If they agree with each other withinthe specified accuracy, the process proceeds to step B5, in which theagreement between the market-dependency of the Boltzmann model and theactual record is checked. When the both values agree with each otherwithin the certain range, the process proceeds to step B2 to determine avolatility matrix. When there is disagreement in step B5, the processreturns to step B3 to reselect the parameters, and to repeat theabove-mentioned procedure. In reality, an apparent market-dependency israrely observed, and therefore, the step B4 may be the finaldetermination to proceed to step B2 (as indicated by the dashed arrow).

Next, how the dealing system 100 evaluates an option price matching withthe historical information, based on the Boltzmann model, will beexplained. It has already been mentioned that the Boltzmann model iscapable of evaluating the smile and the term structure of the impliedvolatility, while keeping consistency with the historical information ofthe underlying assets for reproducing the derivatives. This feature isvery advantageous, because the basis of the option price is clearlyindicated when conducting transaction. In the current status for optionpricing in the real market, rationale for pricing is not necessarilyrequired. Especially when the transaction is carried out within the fullscope of self-responsibility, large losses due to mispricing from thedealer's judgement is attributed to a problem of the person concerned.On the other hand, in the consulting business for transaction or optionpricing, it is required that pricing options must be determined with thereasonable bases, other than long experiences and intuition basis.

Since the uncertainty is high in the market, arbitrary judgment relyingupon experiences and intuitions will never completely disappear.However, if such arbitrary judgment is rationalized by other adequateinformation, the judgement becomes an action based on the rationale, andis not mere manual judgement any longer. The modern financialengineering stands for foundation that pricing derivatives are uniquelydetermined by the price of the underlying asset. Therefore, the matchbetween the Boltzmann model and the historical information of theunderlying assets can be a strong basis of rationality of priceevaluation.

It is recognized that pricing options by the jump model or volatilitymatrix model, previously described, cannot fully cover wider range ofthe historical information. In these cases, it is difficult to explainreasonable sources for the deviation from the reality in the optionmarket. Since such models are not perfect, option pricing needs the newmodel which can reproduce option price from the historical informationfrom the underlying assets. However, the Boltzmann model is advantageousbecause it keeps consistency with the historical information on theunderlying assets in principle with a very little deviation, and inaddition, there are a relatively few occasions that require deviation.Even if such deviation is observed, dealers can discuss with theinterested parties in advance. Therefore, a situation in whichmisjudgment by a particular dealer or a consultant may lead to a hugeamount of losses can be avoided.

The dealing system 100 must evaluate risk parameters expressed byequations (47) through (51) as a risk hedge when evaluating a optionprice. $\begin{matrix}{\Delta = \frac{\partial C}{\partial S}} & (47) \\{\Gamma = \frac{\partial^{2}C}{\partial S^{2}}} & (48) \\{P = \frac{\partial C}{\partial r}} & (49) \\{\Theta = \frac{\partial C}{\partial\tau}} & (50) \\{V = \frac{\partial C}{\partial\sigma}} & (51)\end{matrix}$here C is the option price, S is the underlying price, r is thenon-risky interest rate, τ is the term to the maturity, and σ is thevolatility. It is known that if the underlying assets are traded inproportion to these risk parameters, the price fluctuation of theunderlying assets can be canceled out in principle.

These risk parameters are the differentials of the option price. TheBoltzmann model uses the Monte Carlo method for numerical analysis as aprerequisite; however, the Monte Carlo method has a drawback of takingcomputation time for evaluating the differentials. For instance, whencalculating the risk parameter Θ of a call option price C strictly usingthe Monte Carlo method, an infinitesimal change δτ in term τ to thematurity is set, and equation (52) is computed. $\begin{matrix}\begin{matrix}{\Theta = \frac{\partial C}{\partial\tau}} \\{= \frac{{{Call}\quad( {K,{\tau + {\delta\quad\tau}}} )} - {{Call}\quad( {K,\tau} )}}{\delta\quad\tau}}\end{matrix} & (52)\end{matrix}$

The numerator of equation (52) becomes equation (53) from equation (36).Call(K,τ+δτ)−Call(K,τ)=e ^(−rτ)(e ^(−r·δτ) ∫dS(S−K)P(S,τ+δτ)−∫dS(S−K)P(S,τ)  (53)

The integral of the right-hand side of equation (53) is computed by theMonte Carlo method. The change in τ of the input variable is very small,and therefore, the difference between the integral of the first term andthe integral of the second term of the right-hand side of equation (53)is also small.

Since the computation result of the Monte Carlo method varies within therange of the statistics error, the amount of computation must beincreased in order to reduce the statistic error. In general, thestatistic error is inversely proportional to the square of thecomputation amount. Accordingly, if the change is very small, asignificant difference can not be detected unless a vast of time isspent for computation.

This problem applied to not only the financial Monte Carlo, but also thegeneral Monte Carlo, and an ultimate solution has not been found yet.The neutron transport Monte Carlo method uses a perturbation MonteCarlo, which simulates only an infinitesimal change. However, even withthis method, some approximation is required, and the advantage of theMonte Carlo method, that is, the exactness may be damaged.

Considering the fact that the Monte Carlo method can not conduct arigorous simulation of an infinitesimal change at present, it is notpractical to stick to the exact solution. Under a circumstance in whichthe option price can be explained by the implied volatility, separationbetween the Boltzmann model and the Black-Sholes model is small. Becausethe first-order or the second-order differentials, such as riskparameters, do not depend on models very much, risk parameters can beevaluated at a practically sufficient accuracy by inputting the impliedvolatility calculated back from the option price of the Boltzmann modelinto the Black-Sholes risk parameter evaluation equation.

To be more precise, the volatility σ of risk parameter evaluationequations (54) through (58) based on the Black-Sholes equation isreplaced by the implied volatility that coincides with the option priceof the Boltzmann model. Then, the Black-Sholes risk parameters are usedas those of the Boltzmann model. $\begin{matrix}{\Delta = \{ \begin{matrix}{{N( d_{1} )};{Call}} \\{{{N( d_{1} )} - 1};{Put}}\end{matrix} } & (54) \\{\Theta = \{ \begin{matrix}{{{- \frac{S\quad{Erf}\quad( d_{1} )\sigma}{2\sqrt{\tau}}} - {{rK}\quad{\mathbb{e}}^{{- r}\quad\tau}N\quad( d_{2} )}};{Call}} \\{{{- \frac{S\quad{Erf}\quad( d_{1} )\sigma}{2\sqrt{\tau}}} + {{rK}\quad{\mathbb{e}}^{{- r}\quad\tau}N\quad( {- d_{2}} )}};{Put}}\end{matrix} } & (55) \\{\Gamma = \{ \begin{matrix}{\frac{{Erf}\quad( d_{1} )}{S\quad\sigma\sqrt{\tau}};{Call}} \\{\frac{{Erf}\quad( d_{1} )}{S\quad\sigma\sqrt{\tau}};{Put}}\end{matrix} } & (56) \\{V = \{ \begin{matrix}{{S\sqrt{\tau}{Erf}\quad( d_{\quad 1} )};{Call}} \\{{S\sqrt{\tau}{Erf}\quad( d_{\quad 1} )};{Put}}\end{matrix} } & (57) \\{P = \{ \begin{matrix}{{K\quad\tau\quad{\mathbb{e}}^{{- r}\quad\tau}N\quad( d_{2} )};{Call}} \\{{{- K}\quad\tau\quad{\mathbb{e}}^{{- r}\quad\tau}N\quad( {- d_{2}} )};{Put}}\end{matrix} } & (58)\end{matrix}$here, Erf (x) is an equation defined by $\begin{matrix}{{{Erf}(x)} = {\frac{1}{\sqrt{2\quad\pi}}{{\exp( {{- \frac{1}{2}}x^{2}} )}.}}} & (59)\end{matrix}$

The dealing system 100 may have an alternative function, that is, afunction for generating a table from the probability density functionsevaluated by the Boltzmann model, and for computing the option pricebased on the Riemann sum of vectors, instead of on the recalculation ofthe Monte Carlo method.

Implied volatility is a volatility of the Black-Sholes equation computedbackward so that the option price evaluated by the Boltzmann modelagrees with the option price evaluated by the Black-Sholes (BS)equation. In the Boltzmann model, the option price is expressed byequations (36) and (37), and the integrals of these equations areobtained using the Monte Carlo method.

If the probability density function P (S, τ) does not change largelywith respect to S, that is, if a general price distribution can beapplied, then it is effective and useful to make a table of probabilitydensity functions with respect to S, and to make approximation by theseries of equations 60 and 61. In this case, the approximated resultbecomes quite close to the rigorous evaluation result of the Monte Carlomethod. $\begin{matrix}{{{Call}\quad(K)} \approx {{\mathbb{e}}^{{- r}\quad\tau}{\sum\limits_{{Si} = K}^{{Si}arrow\infty}{\Delta\quad{Si}\quad{P( {{Si},\tau} )}( {{Si} - K} )}}}} & (60) \\{{{Put}\quad(K)} \approx {{\mathbb{e}}^{{- r}\quad\tau}{\sum\limits_{{Si} = 0}^{{Si} = K}{\Delta\quad{Si}\quad{P( {{Si},\tau} )}( {K - {Si}} )}}}} & (61)\end{matrix}$here the probability density function P (Si, τ) is expressed by equation(62), and is evaluated by the Monte Carlo method. $\begin{matrix}{{P( {{Si},\tau} )} = \frac{\int_{{Si} - \frac{1}{2\quad\Delta\quad{Si}}}^{{Si} + \frac{1}{2\quad\Delta\quad{Si}}}{{\mathbb{d}s}\quad{P( {S,\tau} )}}}{\Delta\quad{Si}}} & (62)\end{matrix}$

By storing the probability density functions of equation (62) as a tablewith a practically sufficient small Δsi, equations (60) and (61) can becomputed at a high speed because these two equations are merely the sumof product (or the Riemann sum) of vectors.

Next, the operation process by the dealing system 100 will be explained.FIG. 37 illustrates a sub-screen (or a window) of the dealing terminal105, which displays the detailed track of stock index in a continuoussession. FIG. 38 illustrates a sub-screen of the dealing terminal 105,which displays a table of implied volatility and market prices of eachdelivery month and each exercise price of stock index option togetherwith the stock index as the underlying asset prices.

FIGS. 39 (a) and (b) also illustrate sub-screens of the dealing terminal105, which display the information contained in the table as graphs.FIG. 39(a) is a graph of implied volatility as a function of exerciseprice, which exhibits a so-called smile curve. FIG. 39(b) is a graph ofoption price of each delivery month as a function of exercise price.

This dealing system 100 carries out the operation flow shown in FIG. 40.In the normal market state, the system takes the market data 101 in(step S05), and conducts rough calculation (step S10). The roughcalculation result is displayed on the sub-screen shown in FIG. 37 (stepS15), thereby improving the display speed.

If an abrupt change occurs to the underlying assets as indicated by thedashed circle “a” in FIG. 37, the user can switch the sub-screen (or thewindow) shown in FIG. 41(a) to a sub-screen of detailed informationshown in FIG. 41 (b). To be more precise, the user enlarges the windowby dragging a desired area 111 (corresponding to K4-K5) shown in FIG.41(a) in a diagonal direction using a mouse or other pointing device ina diagonal direction (steps S 20 and S25). Then, additional columns areproduced corresponding to the dragging amount, as shown in FIG. 42(b).The additional input data, that is, virtual exercise prices K451, k452,and K453 are automatically transferred to the BMM 103 for interpolation(steps S30-1 and S30-2).

The interpolation result, that is, the implied volatility and the optionexercise prices corresponding to the abrupt change “a” are returned fromthe BMM 113 to the dealing terminal 105 (step S30-3). Then, the scale ofthe sub-screen of FIGS. 38 and 42(a) and the scale of the sub-screen ofFIGS. 39 and 41(a) are refreshed (step S 35-1) to display the detailedinformation. The interpolated results (i.e., the detailed information)are connected into a line, and displayed on the sub-screen (step S35-2).FIG. 41(b) shows the interpolated detailed implied volatility C65, incomparison with the rough calculation result C64.

Next, the computation carried out by the BMM 103, which is a theoreticalcomputation server, will be explained with reference to FIG. 43.

As has already been mentioned, implied volatility (IV) is a volatilityof the Black-Sholes (BS) equation calculated backward in such a mannerthat the option price evaluated by the Boltzmann model agrees with theoption price provided by the BS equation.

In the Boltzmann model, the option prices of call option and put optionare described by equations (36) and (37), and the integrals of theseequations are obtained using the Monte Carlo method. If the probabilitydensity function P (S, τ) in equations (36) and (37) does not exhibit anextremely big change, that is, if the ordinary price distribution isapplicable to the evaluation, then, the probability density functionwhen it is not largely varied, i.e., when the normal price distributioncan be applied, then the probability density functions P (S, τ) arearranged in a table with respect to S. By making approximation using theseries of equations (60) and (61), a considerably accurate result closeto the strict evaluation result by the Monte Carlo method can beobtained.

Here, the probability density function P (Si, τ) contained in equations(60) and (61) is expressed by equation (62), which is evaluated by theMonte Carlo method. By taking ΔSi sufficiently and practically small,and by storing the probability density function of equation (62) as atable, equations (60) and (61) become merely Riemann sums of vectors,and can be computed at a high speed.

FIG. 43 schematically illustrates the series of equation (60). A smoothcurve C61 in FIG. 43 is the genuine probability density P (S, τ). Thehistogram C62 is the tabled probability density P (Si, τ). The MonteCarlo integral of the product of the genuine probability density P (S,τ) and the linear line C63 with a slope 1 and starting from the point ofS=K is the strict solution of equation (36). If the user requests aninterruption for evaluating a detailed result (YES in step S20 of FIG.40), the rigorous solution of equation (36) is computed. On the otherhand, the integral of the product of the linear line C63 and thehistogram C62 is an approximation expressed by equation (60). In thenormal market state (NO in step S20), approximation is computed as arough evaluation.

In this manner, if detailed evaluation is desired, stricter result isobtained and displayed. In the graph of FIG. 41 (b), a symbol Cindicates the separation between the ordinary rough calculation C64 andthe detailed evaluation C65.

With this arrangement, the evaluation result is displayed promptly basedon the rough calculation in the ordinary state. Under a user request,the window or the sub-screen is enlarged to display the detailedevaluation, thereby allowing the user to visually know the market changequickly.

Next, a technique for developing a structured bond or an exotic optionby the dealing system 100 shown in FIG. 28 will be explained. Such adevelopment can be realized by obtaining and displaying volatilities ofarbitrary multiple terms to evaluate the term structure of thevolatility independent of the market. FIG. 44 illustrates the operationflow of evaluation of multi-term volatility. The same steps as those inthe flowchart of FIG. 40 are denoted by the same numerical references.

In the ordinary state, the dealing terminal 105 displays the market onthe sub-screen based on the rough calculation result, as shown in FIGS.37 through 39. If the user wishes to evaluate volatilities of multipleterms, a term setting window 200 shown in FIG. 45 is called. In order toselect the evaluation period for a selected option, a starting date 201,the maturity 202, and the evaluation interval 203 are input.

The input information does not have to be those values set in themarket. For example, the options in the first and second delivery monthscircled with “a” in FIG. 46, which shows the implied volatility and themarket prices, are traded in the market. However, there is notransaction of the option of the m^(th) delivery month, which is circledwith “b”.

If the user requests an interruption (step S20′ in FIG. 44) forevaluation of the m^(th) delivery month, the user inputs the final dateof the m^(th) delivery month in the item 202 on the screen 200 with theselection of monthly evaluation (step S25′). Then, the inputtedinformation is automatically transferred to the BMM 103 forinterpolation (step S30′; steps S30-1′ and S30-2′) upon hitting “enter”.

The BMM 103 carries out the computation based on Boltzmann model. Thedealing terminal 105 receives the computation result, more specifically,the implied volatility and option price “b” of the m^(th) delivery month(Step S30-3′). Then, the scale of the sub-screen shown in FIGS. 38 and39 are refreshed, and the interpolation results are connected into aline and displayed on the refreshed screen (steps S35-1, S35-2; S35).

FIG. 47 illustrates an example of the interpolation result, in which thecurves C71 and C72 represent the volatility of the option existing inthe market, and a curve C73 represents the volatility of the option thatdoes not exist in the market.

In this manner, obtaining and displaying the volatility of arbitrarymultiple terms allows the term structure of the option volatility thatdoes not exist in the market to be evaluated. This arrangement canimprove the efficiency of developing a structured bond or an exoticoption.

Next, a technique for displaying the behavior of the term structure ofthe implied volatility (IV) at the money (ATM) with fading animationswill be explained with reference to FIGS. 48 through 52. The fadinganimation is also carried out by the dealing system 100.

The flowchart in FIG. 48 includes the fading steps, which are basicallyadded to the operation flow shown in FIG. 40. The same steps as those inFIG. 48 are denoted by the same numerical references. The newly addedfunction is the fading display in step S150.

In the ordinary state, the market is displayed on the sub-screen of thedealing terminal 105, as shown in FIGS. 37 through 39 based on the roughcalculation (step S10). If an abrupt change occurs in the underlyingassets (as indicated by “a” in FIG. 37), the user can select a desiredarea 110 in the sub-screen of FIG. 39 to look into the state. The usercan enlarge the selected area for detailed evaluation, as shown in FIG.41(a), using a mouse (YES in step S20).

In response to this user interruption, the necessary data provided bythe user in the enlarged screen is automatically transferred to the BMMengine 103. The simulation result by the BMM engine 103 is returned tothe dealing terminal 105, and displayed on the sub-screen, as shown inFIG. 50. The column KR shown in FIG. 50 represents virtual real-time ATMinterpolated between the discrete values of the exercise price bands setin the market. The virtual ATM is assumed to be equal to the real-timeunderlying price.

Since the risk index and the option price change most significantly nearATM, it is very important for dealers and traders to observe the marketchange and the term structure in this area.

With the dealing system 100 of the present invention, even if theunderlying price moves between discrete exercise price bands set in themarket, virtual ATM is interpolated as indicated by KR in FIG. 50.Dealers can flexibly evaluate the term structure of the impliedvolatility based on the interpolation.

However, a problem is that the amount of information displayed on thegraph of FIG. 39(b) inevitably increases. For example, if the userwishes to evaluate up to the sixth delivery month, six option pricesmust be displayed on the graph. In this case, it is desirable for theuser to know these evaluation results correctly and promptly at aglance.

To respond to this demand, the behavior of the term structure of theimplied volatility is displayed in fading animations, as shown in FIG.52, based on the operation flow shown in FIG. 49. In FIG. 52(a), thesolid line draws only the implied volatility of the a1^(st) deliverymonth (step S150-4). After a time lapse of a weight level w, the impliedvolatility of the a1^(st) delivery month is displayed by the dashedline, and the implied volatility of the a2^(nd) delivery month isdisplayed by the solid line, as shown in FIG. 52(b) (steps S150-5,S150-6). This step is repeated until the ae^(th) delivery month, asshown in FIG. 52(d) (steps S150-7, S150-8, . . . , S150-e, S150-(e+1)).

The weight level w can be adjusted by typing the plus (+) key or theminus (−) key of the keyboard, as shown in steps S150-1 through S150-3in FIG. 49. In response to the key manipulation, an interruption routineis activated, and this interruption is reflected to the fading displayshown in FIG. 52 at once. The interruption (i.e., adjusting the weightlevel) may be carried out by using a mouse or a pointing device, insteadof keyboarding. The computation result of FIG. 50, which was returnedfrom the BMM engine 103, is display as a graph, as shown in FIG. 51. InFIG. 51, KR denotes the virtual ATM, the white triangles denote theoutput data from the BMM engine 103, and the cross marks denote themarket data.

Displaying the behavior of the term structure of the implied volatility(IV) in ATM by fading animation can protect the user from mispricing dueto too sensitive reaction to the market change.

FIG. 53 is a flowchart including a fading function, which is basicallyadded to the operation flow of FIG. 44. The same steps as those in theflowchart of FIG. 44 are denoted by the same numerical references. Thenewly added step is the display processing of step S150′.

In the flowchart of FIG. 53, the market color associated with marketactivity is displayed on the sub-screen of the dealing terminal 105 inthe ordinary state based on the rough calculation result, as shown inFIGS. 37 through 39 (step S10). If the user wishes to evaluate thevolatility of arbitrary multiple terms simultaneously, the user inputsthe items of staring date 201, the maturity 202, and the evaluationinterval 203 through the term setting page shown in FIG. 45 (step S20′)to set the term of a desired option.

In response to this user interruption, the data inputted by the user isautomatically transferred to the BMM engine 103, and the computationresult shown in FIG. 46 is returned (steps S25′ and S30′). Thiscomputation result is displayed as a graph, as shown in FIG. 47 (stepS35).

If the user designates the fading mode, multi-term volatility isdisplayed on the screen by fading animation based on the operation flowshown in FIG. 49.

By displaying the animated behavior of the term structure of the impliedvolatility of arbitrary multiple terms in a fading manner, undesirablemispricing due to excessive reaction to the market change can beprevented.

Another significant function of the dealing system 100 shown in FIG. 28is to allow dealers to set positions and to automatically order desiredfinancial products. This function will be explained with reference toFIGS. 54 through 56. In FIG. 54, the same steps as those in the flow ofFIG. 40 are denoted by the same numerical references.

In the ordinary state, the sub-screen of the dealing terminal displaysthe market color associated with market activity based on therough-computation as shown in FIGS. 37 through 39 (steps S05 throughS15), and the user can visually refer to the appropriate standard of theadvanced model.

When the user desires to order transactions, the user requests aninterruption at step S40, and drags the areas L, M and N shown in FIG.55 in turn using a mouse (steps S40 and S45). If the user firstdesignates the area L, the transaction input page 210 shown in FIG. 56is started. The page 210 exhibits the attributes of the designatedposition, such as exercise prices, call/put option, implied volatility,and limit price. The user can select in this page purchase or sell,together with the desired number of bills. Upon hitting the “enter” key,the designated order is stored in the order database 130 (step S50). Theother areas M and N are handled in the same manner.

After the order was made, it is checked every t seconds in real timewhether the market has regressed to the target area (steps S60, S10,S15, S20 and S40).

If the market has regressed to the target area (YES in step S60 of FIG.54), the user's order is transferred to the market immediately andautomatically (steps S65 and S70). If the position in the target area(e.g., area N) has been contracted as a trade date (step S76), then areaN is eliminated from the page (step S75).

This operation flow allows the user (or the dealer) to visually checkthe appropriate standard of the advanced model, and to make orderstimely in an automatic manner.

This automatic ordering function based on a selected position may becombined with a fading function. In this case, the behavior of the termstructure of the implied volatility ATM is displayed as a fadinganimation on the dealing terminal 105. FIGS. 57 through 61 illustratethe operation flow of the fading function combined with the automaticordering function. The same steps as those in other flowcharts aredenoted by the same numerical references.

In the ordinary time, the market color is displayed on the sub-screen ofthe dealing terminal 105 based on the rough computation result, as shownin FIGS. 37 through 39 (steps S05, and S10).

Since the risk index and the option price change significantly near ATM,it is very important for dealers and traders to observe the marketchange and the term structure. With the dealing system of the presentinvention, even if the underlying price changes between discreteexercise price bands set in the market on the sub-screen shown in FIG.37, ATM is interpolated as indicated as symbol KR in FIG. 62. The termstructure of the implied volatility is then displayed by fadinganimation (steps S150-1 through S150-(e+1)).

Based on the information displayed on the dealing terminal, the userclicks a desired position (for example, area L) in the displayed page ofFIG. 62 (steps S40, S45, S50). It is automatically checked every tseconds whether the market has regressed to the target area in step S60.If the market has regressed to the target area, transaction order isimmediately transferred the market in an automatic manner (steps S65,S70).

In this manner, the dealing system 100 of the preferred embodimentallows the dealer to visually check the appropriate standard of theadvanced model with a fading animation, and to select a desired positionto make a transaction order timely and automatically. Because thebehavior of the term structure of the implied volatility (IV) in ATM isdisplayed by animation in a fading manner, the dealer can avoidmispricing due to an excessive reaction to the market change.

Next, an automatic warning function of the dealing system 100 will beexplained with reference to FIGS. 63 and 64. In FIG. 63, the same stepsas those in the flowcharts of FIGS. 57 through 61 are denoted by thesame numerical references.

In the ordinary state, the market is displayed on the sub-screen of thedealing terminal 105 based on rough computation, as shown in FIGS. 37through 39 (steps S05, S10, S150), and the user can visually refer theappropriate standard of the advanced model.

In the sub-screen shown in FIG. 64, curve C101 represents the output ofthe Boltzmann model engine 103, curve C102 represents the predictedregression of the market, and curve C 103 represents the impliedvolatility of the market. If the user wishes to make a transaction orderat position “a” in FIG. 64, the user clicks the position to designatethis area for a transaction order in step S80.

At this point of time, the user can select the automatic warningfunction for automatically giving a caution for the selected position“a” in response to the market change. To be more precise, the userrequests an interruption in step S80, and selects a warning area “b” onthe sub-screen shown in FIG. 64 in step S85. The selected warning areacorresponds to the market area for which the user needs warning. Thedesignated warning information is stored in the warning area database131.

If the user selects multiple positions, and wishes to set a warning areafor each position individually, the user repeats the interruption asmany time as the user wishes.

As shown in FIG. 64, the selected ordering position “a” corresponds tothe option of the exercise price K2, and the warning area for thisposition covers both the exercise prices K2 and K3. This is one of thesignificant features of this warning function. The user can flexiblytake exercise prices (i.e., option issues) of other than the designatedposition as a warning condition.

After the warning area has been selected, it is automatically checked instep S90 whether the market has entered the warning area “b”. Thisdetermination is conducted by checking both the warning area database131 and the market database 101 in real time every t seconds. If theprediction of the market fails and the real market represented by curveC104 enters the warning area “b”, then the position “a” at which atransaction order is requested is immediately flickered, as indicated bythe hatched area “d” in FIG. 64 to caution the user (step S95).

This warning function enables a risk manager to carry out appropriaterisk management because a caution is automatically issued when themarket has entered the warning area designated by the dealer. An extrafunction for suggesting an alternative position may be added to theabove-described warning function. In this case, the operation flow shownin FIG. 65 is inserted after step S85 of FIG. 63, as indicated by symbol“D”. If the prediction of the existing ordering position fails, and ifthe real market enters the warning area “b” as indicated by curve C104,then an alternative position area “e” is suggested, as shown in FIG. 66,in addition to giving a caution.

Prior to suggesting the alternative position area, the sub-screen of thedealing system has been displaying a rough computation result, as shownin FIGS. 37 through 39, and an ordering position “a” and thecorresponding warning area “b” are designated on the sub-screen, asshown in FIG. 66.

Now, if a change occurs to the real market, and if the market index C104of the real market enters the warning area “b”, the position “a” isimmediately flickered to give a caution to the user. Then, in step S86shown in FIG. 65, an alternative position area “e” is calculated basedon the ordering data 131 and the market data 101. The alternativeposition area falls into an opposite position for compensating the gapbetween the predicted position and the real market represented by C104.

The alternative position “e” is immediately displayed on the dealingterminal 105 in step S88, as shown in FIG. 66.

The user can again make a transaction order at a desired position “f”inside the alternative position area “e”.

By adding the function for suggesting an alternative position to thewarning function, a vast amount of loss due to an excessive reaction tothe market change can be effectively avoided in advance, because analternative position is automatically extracted together with a caution.

As has been described above, the dealing system of the present inventionhas many significant advantages over the conventional technique based ona general theory of financial engineering, the application of which isvery limited in a non-active option market. The dealing system of thepresent invention has a Boltzmann model computation engine for applyinga theory of nuclear reactor to the financial field, which is capable ofproviding a theoretical price or a risk index meaningful to dealers andtraders through an interactive graphical interface in a flexible manner.Accordingly, the dealing system of the invention can deal with a bigprice change in the underlying assets in a flexible manner.

Although the present invention has been described based on examples ofstock index option as the preferred embodiments, the invention is notlimited to these examples. The present invention is applicable to anyoption products, for example, individual stock options and currencyoptions, the underlying assets of which exhibit a behavior of thegeometric Brownian motions.

1. A dealing system comprising: an implied volatility computation engineconfigured to evaluate an implied volatility based on market data; aBoltzmann model computation engine configured to evaluate an optionprice for a selected option product based on a Boltzmann model using themarket data; a filter configured to convert an option price obtainedfrom the Boltzmann model computation engine into a volatility of aBlack-Sholes equation; and a dealing terminal configured to displayvolatility of the Black-Sholes equation in comparison with impliedvolatility calculated from the market data, or to display the optionprice calculated by the Boltzmann model computation engine in comparisonwith an option price in market.
 2. The dealing system according to claim1, wherein the selected option product is a stock price index option. 3.The dealing system according to claim 1, wherein the selected optionproduct is an individual stock option.
 4. The dealing system accordingto claim 1, wherein the Boltzmann model computation engine has a unitconfigured to calculate an option price consistent to historicalinformation from the market data.
 5. The dealing system according toclaim 1, wherein the Boltzmann model computation engine has a converterconfigured to convert an option price, which was obtained in a discretemanner with respect to an exercise price based on the Boltzmann model,into a volatility obtained from the Black-Sholes equation in order toobtain the option price and risk parameter through interpolation withthe Black-Sholes equation.
 6. The dealing system according to claim 1,wherein the Boltzmann model computation engine has a table generatorthat generates a table of a probability density function evaluated bythe Boltzmann model, and calculate an option price using a sum of innerproducts of vectors.
 7. A computer-readable recording medium storing adealing program for use in a dealing system, the program comprising thesteps of: causing the dealing system to compute an implied volatilityusing market data supplied to the dealing system; causing the dealingsystem to compute an option price based on a Boltzmann model usingmarket data for selected derivatives; causing the dealing system toconvert the option price obtained from the Boltzmann model into avolatility of a Black-Sholes equation; and causing the dealing system todisplay the volatility of the Black-Sholes equation in comparison withimplied volatility calculated from the market data, or to display theoption price based on the Boltzmann model in comparison with an optionprice in a market.